Linear Algebra Highlights


 Emil McDaniel
 1 years ago
 Views:
Transcription
1 Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x a n x n. We can have m equations in n variables, a system of linear equations, which we want to solve simultaneously. Consistent system  has at least one solution; so, has 1 or infinitely many solutions. If it has infinitely many solutions, write them parametrically. Inconsistent system  has no solution. Two systems are equivalent if they have the same solution set. Thus, our goal is to get a system into a simpler form which is equivalent to the original using the Allowable Operations. A m n matrix is a rectangular array of real numbers with m rows and n columns. Entries are labeled a ij where i is the row the entry is in and j the column. We can represent a system of equations in matrix form: A x = b. This is easier to work with, plus matrices are interesting in their own right. Elementary Row Operations (EROs)  1. Interchange rows R i R j 2. multiply a row by a nonzero constant cr i 3. add a multiple of one row to another R i + cr j (replaces row i). Two matrices are row equivalent if you can obtain one from the other using EROs. Row equivalence satisfies the properties: reflexive (A is row equivalent to itself  do nothing), symmetric (if A is equivalent to B, B is equivalent to B  can get back to the original), and transitive (if A is equivalent to B and B to C, then A is equivalent to C  as you are using the EROs all intermediary steps are also row equivalent). Namely, REF and RREF are row equivalent to the original, so we can use those forms to answer questions about the original matrix, such as finding solutions to systems of equations. Row Echelon Form (REF)  all rows consisting entirely of 0 s are at the bottom of the matrix, 1st nonzero entry in each row is a 1, leading 1 in a higher row is farther left than the leading 1 in a lower row. If you put it in this form, use back substitution to find the answer (solving system this way is called Gaussian elimination with back sub). Reduced Row Echelon Form (RREF)  REF with the additional requirement that every column containing a leading 1 has 0 s in the rest of the column. Can read off the answer in this form (usually preferred); putting a matrix into RREF is called GaussJordan elimination. We can always put a matrix in these forms. The RREF of a matrix is unique (the REF is not). When solving a system of equations using an augmented matrix, a row/rows of all zeros mean there are infinitely many solutions. The column(s) missing a leading 1 is/are the free variable(s) (use a different parameter letter for each free variable). If you have a row of zeros equaling a nonzero constant term, there are no solutions. If the coefficient matrix is row equivalent to the identity matrix, there is exactly 1 solution. Homogeneous system  all constant terms are 0. Always consistent: x 1 = = x n = 0 is a solution. 1
2 An underdetermined system has more variables than equations; it usually (but not always) has infinitely many solutions (it never has 1). An overdetermined system has more equations than variables; it usually is inconsistent (but not always). Chapter 2 Let A = [a ij ], B = [b ij ], C = [c ij ] and c be a constant. Let I be the identity matrix (1 s on diagonal, 0 s elsewhere) and [0] be the matrix with all entries equal to 0. A + B  add matrices by adding corresponding entries: A + B = [a ij + b ij ]. Matrices must be the same size in order to add them. ca  scalar multiplication  multiply every entry in the matrix by the scalar: ca = [ca ij ]. AB  the number of columns in the first matrix must equal the number of rows in the second; if A has size m n and B has size n p, then AB has size m p. The ij th entry of AB is n the dot product of the i th row of A with the j th column of B: c ij = a ik b kj. In general matrix multiplication is not commutative: AB BA. AB = BA for all B. k=1 A = ci if and only if Diagonal matrix  square matrix where all nondiagonal entries are 0; a ij = 0 whenever i j. Trace  the trace of a square matrix is the sum of its diagonal entries: T r(a) = a 11 + a a nn. It is a linear function: T r(a + B) = T r(a) + T r(b) and T r(ca) = ct r(a). The set of all n n matrices over real numbers is an algebra. The following properties hold: 1. there is an additive identity: A + [0] = [0] + A = A 2. every matrix has an additive inverse: A + ( A) = A + A = [0] 3. additive associativity: A + (B + C) = (A + B) + C 4. additive commutativity: A + B = B + A 5. scalar distributes: c(a + B) = ca + cb 6. matrix distributes over scalars: (c + d)a = ca + da 7. scalar associativity: (cd)a = c(da) 8. scalar multiplicative identity: 1A = A 9. c(ab) = (ca)b = A(cB) 10. distributivity: A(B + C) = AB + AC and (A + B)C = AC + AB 11. multiplicative associativity: A(BC) = (AB)C 12. multiplicative identity: AI = IA = A. Note: all these properties hold for any size matrices as long as the sizes are such that the sums and products are defined. 2
3 Be careful of the following: AB does not necessarily equal BA (see above). We do not in general have the cancellation law: AC = BC does not imply A = B in general. AC = BC or CA = CB imply A = B if C is invertible. AB = [0] does not imply that A = [0] or B = [0]. laws of exponents hold for square matrices Transpose  the transpose of a matrix A is switching the columns to rows and rows to columns. If A is size m n, then A T is n m. Form A T by a ij a ji for all i, j. Notice that the diagonal entries remain the same. Properties of transpose: (A T ) T = A, (A + B) T = A T + B T, (ca) T = c(a T ), (AB) T = B T A T. Especially note the last one. A is symmetric if A = A T. Thus, A must be square. a ij = a ji for all i, j. A is skewsymmetric if A = A T. AA T is symmetric for all matrices. An n n matrix A is invertible (nonsingular) if there exists a matrix B such that AB = I = BA. An inverse of a matrix is unique, denote it by A 1. Not all matrices have inverses. Find the inverse of a matrix by adjoining the identity matrix and use GaussJordan elimination to turn A into the identity matrix and the right side will be the inverse: [A I] [I A 1 ]. If you cannot use the EROs to make the left side the identity, the matrix is not invertible (is singular). Check your answer. For a 2 [ 2 matrix, ] we have a shortcut [ formula (we ] will look at extending this formula later). a b If A =, then A c d 1 = 1 d b ad bc. Note that this means a 2 2 matrix is c a invertible iff ad bc 0. Properties of inverses: 1. (A 1 ) 1 = A 2. (A k ) 1 = (A 1 ) k 3. (ca) 1 = 1/cA 1 4. (A T ) 1 = (A 1 ) T 5. (AB) 1 = B 1 A 1. Note that 1 and 5 are the same as for transpose. Note: the product of two invertible matrices is invertible, however, the sum of two invertible matrices is not necessarily invertible. If A is invertible, then A x = b has a unique solution: x = A 1 b. It is the same amount of work to solve a system this way. The advantage is that if we have trying to solve the same coefficient system with different constants, I only have to solve it once. Idempotent  a matrix A such that A 2 = A. Nilpotent  a matrix A such that A k = [0] for some natural number k. Elementary matrix  a matrix which can be obtained from the identity matrix by performing one ERO. Its inverse exists and is an elementary matrix. Uses: can use matrix multiplication to perform EROs, some results are easier to prove with elementary matrices, used to find LUfactorization. Performing an ERO on a matrix A is the same as multiplying on the left by the elementary matrix corresponding to that ERO. Example: doing R 2 2R 1 to A is the same as forming E by R 2 2R 1 to the identity matrix and then multiplying EA. 3
4 B is rowequivalent to A if there exists elementary matrices E 1,..., E k such that B = E k E 1 A. Thus, B is invertible if and only if it can be written as a product of elementary matrices (A = I). Summary of invertibility equivalences: If A is an n n matrix, then the following are equivalent (all true or all false): 1. A is invertible 2. A x = b has a unique solution for every column matrix b 3. A x = [0] has only the trivial solution 4. A is rowequivalent to I n 5. A can be written as a product of elementary matrices. Lower triangular matrix  all entries above main diagonal are zero. Strictly lower triangular  all entries on and above main diagonal are zero. Upper triangular matrix  all entries below main diagonal are zero. Strictly upper triangular  all entries on and below main diagonal are zero. LUfactorization  writing a square matrix as a product of a lower and upper triangular matrix. Used in an efficient algorithm for solving systems of linear equations. To find the LUfactorization, use EROs to put A into an upper triangular matrix: U = E k E 1 A. Then L = E1 1 E 1 k. If we can use only adding multiples of rows, A definitely has an LUfactorization and it is easy to get L. L will have 1 s on the diagonal and the negative of each multiplier used to obtain U in the same position. To solve a system A x = b using LU, solve L y = b and then solve U x = y. relatively easy since they are in triangular form. Each one is Chapter 4 Let V be a set on which two operations (vector addition and scalar multiplication) are defined. V is a vector space if the following axioms hold for all u,v,w in V and every scalar c,d. 1. closure under vector operation: u v V 2. vector operation is commutative 3. vector operation is associative 4. there exists an vector identity: u æ = u 5. every vector has an inverse 6. closure under scalar operation: c u V 7. a scalar distributes over vectors 8. a vector distributes over scalars 9. scalar operation is associative: (cd)u = c(du) 10. there is a scalar identity: e u = u Standard examples of vector spaces: R n for n 1, m n matrices, P n  polynomials of degree less than or equal to n, realvalued continuous functions Nonexamples of vector spaces: integers, polynomials of exactly degree n, R 2 with scalar operation defined to be c(x, y) = (cx, 0) 4
5 Need to say what the set of vectors, set of scalars, vector operation and scalar operation are. To prove something is a vector space, need to show all 10 axioms hold. To show something is not a vector space, you just need to show one axiom does not hold. vector identity is unique and each vector has a unique inverse A nonempty subset W of a vector space V is a subspace of V if W is a vector space under the same operations as defined in V. It is enough to check 1. nonempty 2. closure under vector operation 3. closure under scalar operation. Note that the vector identity must be in W and will be the same one as in V. Every vector space has at least two subspaces: {æ}  trivial subspace of just the vector identity, V  the entire vector space. Subspaces of R 2 : the two trivial subspaces and any line through (0, 0). Subspaces of R 2 : the two trivial subspaces, any line through (0, 0), plane through (0, 0). Subspaces of M n : symmetric, triangular, many more Look through the exercises in the book to get many examples and nonexamples of vector spaces. The intersection of two subspaces of a vector space U is a subspace (has at least the vector identity). The union of two subspaces of a vector space U is not necessarily a subspace. Some important examples of subspaces of vector space U. vector identity U For subspaces V, W : V W The sum of two subspaces: V + W = {v + w} set of all linear combinations of a set of vectors in U: {c 1 u 1 + c 2 u c k u k } (span) Let V be a vector space. v V is a linear combination of vectors u 1, u 2,..., u k V if v = c 1 u 1 + c 2 u c k u k for some scalars. We can write this as a system of equations to solve for the scalars  the vectors go as columns in a matrix. It can be written as a linear combination as long as the matrix is consistent. Let S = {v 1,..., v k } be a subset of a vector space V. S is a spanning set of V if every vector in V can be written as a linear combination of vectors in S. Say S spans V (span(s)=v). A vector space can have many spanning sets. 5
6 Let S = {v 1,..., v k } be a set of vectors in a vector space V. span(s) = {c 1 v 1 + c 2 v c k v k }  it is the set of all linear combinations of the vectors in S. span(s) is a subspace of V  essentially we take the vectors in S and put in all other vectors necessary to make a subspace. It is the smallest subspace of V which contains S. Let S = {v 1,..., v k } be a set of vectors in a vector space V. The set is linearly independent if the equation c 1 v 1 + c 2 v c k v k = 0 has only the trivial solution c 1 = c k = 0. Else it is linearly dependent. The set is linearly dependent if one of the vectors v i can be written as a linear combination of the others. To test for linear independence: write the system of equations in a matrix and solve. Example: In R n, n vectors  put the vectors as columns in a matrix and they will be LI if the matrix is invertible. A set of vectors S = {v 1,..., v n } in a vector space V is a basis if 1. S spans V and 2. S is linearly independent. You can think of it as a minimal spanning set. Bases are not unique. S does not have to be a finite set. [ ] [ ] Standard bases: R 3 : {(1, 0, 0), (0, 1, 0), (0, 0, 1)}; P 2 : {1, x, x 2 }; M 2 :,, [ ] [ ] , While bases are not unique, every basis for a given vector space V has the same number of vectors. So if a set of n vectors is a basis for V, every basis for V has n vectors. This number, the number of vectors in a basis for V, is called the dimension of the vector space. Examples: The trivial vector space has dimension 0, R n has dimension n, dim(p n )=n+1, dim(m m,n ) = mn. Every vector in V can be written uniquely has a linear combination of vectors in a basis for V. Let V be a vector space of dimension n. Then 1. If S is a LI set of n vectors, then S is a basis. 2. If S is a spanning set of n vectors, then S is a basis. In other words, if I already know the dimension of a vector space, I only have to make sure I have the correct number of vectors and check one of the 2 conditions for a basis. Let A be an m n matrix. The row space of A is the subspace of R n spanned by the row vectors of A. The column space of A is the subspace of R n spanned by the column vectors of A. If two matrices are rowequivalent, then their row spaces are equal. Thus, we can put a matrix A into rowechelon form (or RREF), and the nonzero rows will form a basis for the row space of A. To find a basis for a subspace of R n spanned by a given set of vectors we can place the vectors as rows in a matrix and find the basis for the row space. 6
7 To find a basis for the column space of A, use one of the following methods. 1. column space of A = row space of A T. 2. row operations change the column space, but not the linear dependency, so choose the columns in A with leading ones in RREF(A) 3. use elementary column operations The row space and column space of A have the same dimension, this is called the rank of A = rank(a). This is the number of nonzero rows in REF(A). Let A be a fixed m n matrix. Then {x R n : Ax = 0} is a subspace of R n, called the nullspace of A. The dimension of the nullspace of A is called the nullity of A. To find the nullspace of a matrix A, find RREF(A). Solve the homogeneous system of equations. The number of free variables is the nullity of A. To find a basis, choose values for your free variables, or write the solutions as sx+ty etc; x,y are basis vectors. Thus, the nullity of an invertible matrix is 0. RankNullity Theorem: If A is an m n matrix, then rank(a)+nullity(a)=n. The system of linear equations Ax = b is consistent if and only if b is in the column space of A. Let B = { v 1,..., v n } be an ordered basis (notice the order I write the vectors matter) for a vector space V and x be a vector in V such that x = c 1 v c n v n. The scalars c 1,..., c n are called the coordinates of x relative to the basis B. The coordinate matrix/vector of x relative to B is the column matrix whose components are the coordinates of x. Change of basis  given coordinates of a vector relative to one basis, and want to find the coordinates relative to another basis. To do a change of basis from B to B, find [x] B = P [x] B where P is the transition matrix. To find P, find [B : B] [I : P ] Chapter 5 Let u, v, w be vectors in a vector space V, let c be a scalar. An inner product on V is a function that associates a real number u, v with each pair of vectors and satisfies: 1. u, v = v, u 2. c u, v = c u, v = u, c v 3. u, v + w = u, v + u, w 4. u, u 0 and = 0 iff u =add id. NB: for R n, the standard inner product is the dot product. NB: we need to check 5 things, that it outputs a real number and satisfies 14. A vector space with an inner product is called an inner product space. Inner products on R n are of the form: u, v = c 1 u 1 v c n u n v n where all of the c i > 0. Let u,v be in an inner product space V. The norm of u is u = u, u. The distance between u,v is d( u, v) = u v. The angle between u,v is cos θ = u, u u v. 7
8 Two vectors are orthogonal if their inner product is 0. A unit vector has norm equal to 1. A unit vector in the same direction as v is u = Let u, v be vectors in an inner product space V. CauchySchwarz Inequality: u,v u v Traingle Inequality: u + v u + v v v. Pythagorean Theorem: u and v are orthogonal if and only if u + v 2 = u 2 + v 2 Let u and v be vectors in an inner product space such that v 0. Then the orthogonal u, v projection of u unto v is proj v u = v, v v A set S of vectors in an inner product space V is called orthogonal if every pair of vectors in S is orthogonal. If each vector is a unit vector, then S is orthonormal. We really like orthonormal bases because they are easier to work with. The standard bases are orthonormal. B = {(cos θ, sin θ, 0), ( sin θ, cos θ, 0), (0, 0, 1)} is also an orthonormal basis for R n. An orthogonal set of nonzero vectors in an inner product space is linearly independent. If w is orthogonal to every vector in a set S, then it is orthogonal to every linear combination of vectors in S. If B = { v 1, v 2,..., v n } is an orthonormal basis for an inner product space V, then the coordinate representation of a vector w with respect to B is: w = w, v 1 v 1 + w, v 2 v w, v n v n. Hence, it is easier to find the coordinates in regards to an orthonormal basis than one which is not. GramSchmidt Orthonormalization Process: turns a basis into an orthonormal basis. It does so by using the projection of the vectors to get an orthogonal set, then making them unit vectors. You can make each vector a unit vector as you go along, or make them all unit vectors at the very end. Let B = { v 1, v 2,..., v n } be a basis for an inner product space V. Define w 1 = v 1 w 1 = v 2 v 2,w 1 w 1,w 1 w 1 = v 2 proj w1 v 2... w n = v n proj w1 v n proj w2 v n proj wn 1 v n Then {w 1,..., w n } is an orthogonal basis. Make an orthonormal basis by taking u i = w i w i. An invertible n n matrix P is an orthogonal matrix if P 1 = P T. In other words, P P T = I. The row (column) vectors of P form an orthonormal basis for R n. Let W be a subspace of the inner product space V (W is also an inner product space.) Then the orthogonal complement of W, W, is also a subspace of V. W = {v V : v, w = 0 w W } (read W perp ). It is the set of all vectors orthogonal to all the vectors in W. (W ) = W 8
9 Let U, W be subspaces of a vector space V. If V = U + W and U W = {add id}, then V is the direct sum of U and W : V = U W. Furthermore, every vector in V has a unique representation of the form u + w. V = W W. Thus, dim(w ) + dim(w ) = dim(v ). To find the orthogonal complement of a subspace S of R n, set up the homogeneous system of linear equations of the inner product of each basis vector with a generic vector (v 1, v 2,..., v n ). Then S =nullspace(a); the nullspace of that system. For the dot product on R n, put the basis vectors of S as the rows in a matrix and find the nullspace. Important examples: Let A be an m n matrix. Then R m = row(a) null(a T ) and R n = row(a T ) null(a). Chapter 6 Let T : V W be a function that maps a vector space V to a vector space W. T : V W. V is the domain, W is the codomain, T (V ) is the range. Note that codomain and range have different meanings. T (v) = w. w is the image of v under T. Range is the set of all images of vectors in V. {v : T (v) = w} is the preimage of w. T : V W is a linear transformation of V into W if the following is true for all u, v V and scalars c. 1. T ( u + V v) = T ( u) + W T ( v) 2. T (c V u) = c W T ( u) So, it is a function which preserves vector addition and scalar multiplication. We always have two linear transformations: send everything to the additive identity and the identity transformation. T ( u) =add. id u T : V V  T ( u) = u u Properties of linear transformations 1. T (æ V ) = æ W additive identity goes to additive identity 2. additive inverse goes to additive inverse (c=1 above) 3. T (c 1 v c n v n ) = c 1 T (v 1 ) + + c n T (v n ) linearity extends These properties can help spot functions which are not linear transformations. Also, the 3rd property tells us that a linear transformation is determined by its action on a basis. So, we just have to know what it does to a basis of V to know what it does to every vector in V. Let A be a m n matrix. T ( v) = A v is a linear transformation from R n to R m. Every linear transformation from R n to R m can be represented in this way. Examples of linear transformations: rotation (use sin, cos), projection (column of 0 s), T (A) = A T transpose, differentiation, line through the origin 9
10 A fixed point is a vector which gets mapped to itself under a linear transformation: T : V V, T (u) = u. Note in the identity transformation, all points are fixed. The additive identity is always fixed. kernel of T : V W is {v V : T (v) = æ W } = ker(t). The set of all vectors in V which go to the additive identity in W. The kernel is never empty since the additive identity of V will always be in it. Note: this is an analogue to nullspace since for T (v) = Av the ker(t)=null(a). The kernel of the zero transformation is V and the kernel of the identity transformation is the trivial subspace. The ker(t) is a subspace of V. The dimension of ker(t) is the nullity. range(t)= {T (v) : v V } is a subspace of W. Dimension of range(t) is the rank. For T (v) = Av the range(t)=col(a). We have the RankNullity Theorem for linear transformations: dim(range)+dim(kernel)=dim(domain). A function is onetoone (11) if and only if T (u) = T (v) u = v. (Two different elements do NOT map to the same thing.) T : V W is onetoone if and only if ker(t) is trivial (only contains add. id.) A function is onto if and only if for all w W, there exists v V such that T (v) = w. In other words, the range is W; and thus, rank(t)=dim(w). If V and W have the same dimension, then T is 11 iff T is onto. A linear transformation T : V W which is 11 and onto is called an isomorphism. V and W are isomorphic if there exists an isomorphism T : V W. Isomorphism says that two spaces are essentially the same. They are the same dim, rank, isomorphic subspaces, etc. It says if we understand one vector space, we understand all of them to which it is isomorphic. Let V and W be finitedimensional vector spaces. Then V and W are isomorphic if and only if they have the same dimension. The linear transformation which does this takes v i w i for each basis vector. Note, this is only for finitedimensional and is special to vector spaces (so don t try to do it with groups in Abstract Alg ;)) Every linear transformation can be represented by a matrix: T (v) = Av for some matrix A. Where the basis vectors get mapped to determines the matrix. Let T : R n R m, where both vector spaces have the stand bases. Let e i be the standard basis vector with a 1 in the ith position and 0 s elsewhere. Then T can be defined by T (v) = Av for a m n matrix A. If T (e i ) = [a 1i a 2i... a mi ] T for all i, then A = [a ij ]. Call A the standard matrix. In other words, determine where each standard basis vector goes to, and place those vectors as columns in your matrix A. 10
11 Let T 1 : R n R m and T 2 : R m R p have standard matrices A 1 and A 2. The composition T 2 (T 1 (v)) has standard matrix A 2 A 1. T : V V is invertible if there exists a linear transformation T 1 : V V such that T (T 1 (v)) = T 1 (T (v)) = v. I can get back to the element I started with; compose to get the identity transformation. If the standard matrix for T is A, then the standard matrix for T 1 is A 1. Let T : V V with standard matrix A. The following are equivalent: 1. T is invertible 2. T is an isomorphism 3. A is invertible. If you are not using the standard bases, then you need to find T (v i ) and then do a change of basis using the technique from chap 4. Let T : V W where V has basis B = {v 1,..., v n } and W has basis B. Find [T (v i )] B ; those vectors are the columns in the matrix A relative to bases B and B. Let A, B be n n matrices. A is similar to B if there exists an invertible matrix P such that A = P 1 BP. Write A B. Similarity satisfies reflexive (A is similar to A), symmetric (if A is similar to B, then B is similar to A), and transitive (if A is similar to B and B to C, then A is to C). Chapter 3 The determinant is a map from square matrices to the real numbers. det:m n,n R. It is defined recursively. If A is 2 by 2, det(a)=a 11 a 22 a 21 a 12. To find the determinant of a larger matrix, pick a row or column. For each element in that row/column, multiply the number by the determinant of the smaller matrix obtained from covering up the row and column of that element; signs alternate. The minor M ij of the element a ij is the determinant obtained by deleting the ith row and jth column of A. The cofactor is C ij = ( 1) i+j M ij. n det(a) = A = a ij C ij for any fixed i. You can also switch the i s and j s and use one j=1 column instead. Choose which row/column you use wisely. The determinant of a diagonal or triangular matrix is the product of the diagonal entries. If any row or column is all 0 s, the determinant is 0. Likewise, if any 2 rows are the same or multiples of each other, the det is 0. You can use elementary row operations to find the determinant of a matrix. It is easier this way since the more 0 s you have the easier the determinant. And, EROs are easier than the recursive determinant. If you put the matrix in REF, the determinant will be the product of the diagonal entries. 1. Interchanging two rows changes the sign of the determinant. 11
12 2. Adding a multiple of one row to another does not change the determinant. 3. Multiplying a row by a nonzero constant multiplies the determinant by that constant. Typically by just using (2) we can simplify the matrix enough to find the determinant easily. We do not really have to get 1 s on the diagonal. You can also do column operations. It makes the same changes as above. If A and B are n n matrices, det(ab)=det(a)det(b) and det(ca)= c n det(a). A square matrix is invertible if and only if det(a) 0. det(a 1 ) = det(a)=det(a T ) The determinant of an orthogonal matrix is ±1. If A B, then det(a)=det(b). 1 det(a). The adjoint of A is the transpose of the matrix of cofactors. Each entry a ij in the adjoint matrix is the cofactor of a ji. A 1 = 1 det(a) adj(a). Cramer s Rule  If a system of n equations in n variables has a coefficient matrix with nonzero determinant, then the solution to the system of equations is x 1 = A 1 A,..., x n = An A, where A i is the matrix A with replacing the ith column with the column of constants. Chapter 7 Let A be a square matrix. The scalar λ is an eigenvalue of A if there exists a nonzero vector x such that Ax = λx. x is an eigenvector of A corresponding to λ The eigenspace of λ is the set of all eigenvectors of λ along with the zero vector. It is a subspace of R n. Ax = λx Ax λx = 0 (A λi)x = 0. This system has nonzero (remember eigenvectors cannot be the zero vector) solutions if and only if the coefficient matrix A λi is not invertible if and only if the det(a λi) = 0. We find eigenvalues by setting the polynomial det(a λi) equal to 0 (it has degree n, so there are at most n eigenvalues). We find each eigenvalue s corresponding eigenvectors by solving the system of equations (A λi) x = 0. Finding a basis for the solution spaces gives us a basis for the eigenspace. det(a λi) is called the characteristic polynomial of A and det(a λi) = 0 is the characteristic equation. If λ is a multiple root of order k of the characteristic polynomial, then we say it has multiplicity k. If A is a square triangular matrix, then its eigenvalues are the entries on the diagonal. This is because the det is the product of diagonal entries. 12
13 We can also talk about the eigenvalues/vectors of a linear transformation (Ax = λx is just a special case). An eigenvalue for a linear transformation is a scalar λ such that there exists a nonzero vector x such that T (x) = λx. Eigenvectors and eigenspaces are likewise defined. trace(a) equals the sum of the eigenvalues. det(a) equals the product of the eigenvalues. CayleyHamilton Theorem  a matrix satisfies its characteristic equation; i.e. if p(λ) = 0 is the characteristic equation, then p(a) = 0. A square matrix A is diagonalizable iff A is similar to a diagonal matrix. If A and B are similar matrices, then they have the same eigenvalues. Thus, if A is diagonalizable, it is similar to a diagonal matrix with A s eigenvalues on the diagonal. A is diagonalizable if and only if it has n linearly independent eigenvectors. In other words, the dimensions of all the eigenspaces add up to n. A is diagonalizable if and only if there exists a matrix P such that A = P DP 1 where D is a diagonal matrix with the eigenvalues of A on the diagonal and the columns of P are n linearly independent eigenvectors of A (the basis vectors for all the eigenspaces). So, we get P using the eigenvectors and D using eigenvalues. Distinct eigenvalues have linearly independent eigenvectors, so if A has n distinct eigenvalues, it is diagonalizable. By using the standard matrix for a linear transformation, we can find a basis for V such that the matrix for T relative to B is diagonal. The basis is the eigenvectors of the standard matrix. If A is not diagonalizable it is still similar to an upper triangular matrix with the eigenvalues on the diagonal and 1 s on the superdiagonal corresponding to which eigenspaces do not have full dimension. Real Spectral Theorem  If A is a symmetric matrix, then 1. A is diagonalizable 2. all of its eigenvalues are real 3. if λ has multiplicity k then its eigenspace has dimension k. The matrix P that diagonalizes a symmetric matrix (D = P AP 1 ) is an orthogonal matrix. A matrix A is orthogonally diagonalizable if there exists a diagonal matrix P such that D = P AP 1. Fundamental Theorem of Symmetric Matrices  A is orthogonally diagonalizable and has real eigenvalues if and only if A is symmetric. The End! 13
Math 4A Notes. Written by Victoria Kala Last updated June 11, 2017
Math 4A Notes Written by Victoria Kala vtkala@math.ucsb.edu Last updated June 11, 2017 Systems of Linear Equations A linear equation is an equation that can be written in the form a 1 x 1 + a 2 x 2 +...
More informationMATH 240 Spring, Chapter 1: Linear Equations and Matrices
MATH 240 Spring, 2006 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 8th Ed. Sections 1.1 1.6, 2.1 2.2, 3.2 3.8, 4.3 4.5, 5.1 5.3, 5.5, 6.1 6.5, 7.1 7.2, 7.4 DEFINITIONS Chapter 1: Linear
More informationEquality: Two matrices A and B are equal, i.e., A = B if A and B have the same order and the entries of A and B are the same.
Introduction Matrix Operations Matrix: An m n matrix A is an mbyn array of scalars from a field (for example real numbers) of the form a a a n a a a n A a m a m a mn The order (or size) of A is m n (read
More informationLinear Algebra. Matrices Operations. Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0.
Matrices Operations Linear Algebra Consider, for example, a system of equations such as x + 2y z + 4w = 0, 3x 4y + 2z 6w = 0, x 3y 2z + w = 0 The rectangular array 1 2 1 4 3 4 2 6 1 3 2 1 in which the
More informationGlossary of Linear Algebra Terms. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Glossary of Linear Algebra Terms Basis (for a subspace) A linearly independent set of vectors that spans the space Basic Variable A variable in a linear system that corresponds to a pivot column in the
More informationFinal Review Sheet. B = (1, 1 + 3x, 1 + x 2 ) then 2 + 3x + 6x 2
Final Review Sheet The final will cover Sections Chapters 1,2,3 and 4, as well as sections 5.15.4, 6.16.2 and 7.17.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch
More informationDaily Update. Math 290: Elementary Linear Algebra Fall 2018
Daily Update Math 90: Elementary Linear Algebra Fall 08 Lecture 7: Tuesday, December 4 After reviewing the definitions of a linear transformation, and the kernel and range of a linear transformation, we
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW SPENCER BECKERKAHN Basic Definitions Domain and Codomain. Let f : X Y be any function. This notation means that X is the domain of f and Y is the codomain of f. This means that for
More informationLecture Summaries for Linear Algebra M51A
These lecture summaries may also be viewed online by clicking the L icon at the top right of any lecture screen. Lecture Summaries for Linear Algebra M51A refers to the section in the textbook. Lecture
More informationMath Linear Algebra Final Exam Review Sheet
Math 151 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a componentwise operation. Two vectors v and w may be added together as long as they contain the same number n of
More informationELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS. 1. Linear Equations and Matrices
ELEMENTARY LINEAR ALGEBRA WITH APPLICATIONS KOLMAN & HILL NOTES BY OTTO MUTZBAUER 11 Systems of Linear Equations 1 Linear Equations and Matrices Numbers in our context are either real numbers or complex
More informationLecture Notes in Linear Algebra
Lecture Notes in Linear Algebra Dr. Abdullah AlAzemi Mathematics Department Kuwait University February 4, 2017 Contents 1 Linear Equations and Matrices 1 1.2 Matrices............................................
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationOHSx XM511 Linear Algebra: Solutions to Online True/False Exercises
This document gives the solutions to all of the online exercises for OHSx XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Answers are in square brackets [. Lecture 02 ( 1.1)
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationLinear Algebra March 16, 2019
Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationOnline Exercises for Linear Algebra XM511
This document lists the online exercises for XM511. The section ( ) numbers refer to the textbook. TYPE I are True/False. Lecture 02 ( 1.1) Online Exercises for Linear Algebra XM511 1) The matrix [3 2
More informationSUMMARY OF MATH 1600
SUMMARY OF MATH 1600 Note: The following list is intended as a study guide for the final exam. It is a continuation of the study guide for the midterm. It does not claim to be a comprehensive list. You
More informationMath Camp Lecture 4: Linear Algebra. Xiao Yu Wang. Aug 2010 MIT. Xiao Yu Wang (MIT) Math Camp /10 1 / 88
Math Camp 2010 Lecture 4: Linear Algebra Xiao Yu Wang MIT Aug 2010 Xiao Yu Wang (MIT) Math Camp 2010 08/10 1 / 88 Linear Algebra Game Plan Vector Spaces Linear Transformations and Matrices Determinant
More information1 9/5 Matrices, vectors, and their applications
1 9/5 Matrices, vectors, and their applications Algebra: study of objects and operations on them. Linear algebra: object: matrices and vectors. operations: addition, multiplication etc. Algorithms/Geometric
More informationANSWERS (5 points) Let A be a 2 2 matrix such that A =. Compute A. 2
MATH 7 Final Exam Sample Problems Spring 7 ANSWERS ) ) ). 5 points) Let A be a matrix such that A =. Compute A. ) A = A ) = ) = ). 5 points) State ) the definition of norm, ) the CauchySchwartz inequality
More informationThe value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver I.N.
Math 410 Homework Problems In the following pages you will find all of the homework problems for the semester. Homework should be written out neatly and stapled and turned in at the beginning of class
More informationDefinitions for Quizzes
Definitions for Quizzes Italicized text (or something close to it) will be given to you. Plain text is (an example of) what you should write as a definition. [Bracketed text will not be given, nor does
More informationLinear Algebra: Lecture notes from Kolman and Hill 9th edition.
Linear Algebra: Lecture notes from Kolman and Hill 9th edition Taylan Şengül March 20, 2019 Please let me know of any mistakes in these notes Contents Week 1 1 11 Systems of Linear Equations 1 12 Matrices
More information[Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty.]
Math 43 Review Notes [Disclaimer: This is not a complete list of everything you need to know, just some of the topics that gave people difficulty Dot Product If v (v, v, v 3 and w (w, w, w 3, then the
More informationLinear Algebra M1  FIB. Contents: 5. Matrices, systems of linear equations and determinants 6. Vector space 7. Linear maps 8.
Linear Algebra M1  FIB Contents: 5 Matrices, systems of linear equations and determinants 6 Vector space 7 Linear maps 8 Diagonalization Anna de Mier Montserrat Maureso Dept Matemàtica Aplicada II Translation:
More informationMAT Linear Algebra Collection of sample exams
MAT 342  Linear Algebra Collection of sample exams Ax. (0 pts Give the precise definition of the row echelon form. 2. ( 0 pts After performing row reductions on the augmented matrix for a certain system
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationThe definition of a vector space (V, +, )
The definition of a vector space (V, +, ) 1. For any u and v in V, u + v is also in V. 2. For any u and v in V, u + v = v + u. 3. For any u, v, w in V, u + ( v + w) = ( u + v) + w. 4. There is an element
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationMath 314/814 Topics for first exam
Chapter 2: Systems of linear equations Math 314/814 Topics for first exam Some examples Systems of linear equations: 2x 3y z = 6 3x + 2y + z = 7 Goal: find simultaneous solutions: all x, y, z satisfying
More informationSpring 2014 Math 272 Final Exam Review Sheet
Spring 2014 Math 272 Final Exam Review Sheet You will not be allowed use of a calculator or any other device other than your pencil or pen and some scratch paper. Notes are also not allowed. In kindness
More informationNOTES on LINEAR ALGEBRA 1
School of Economics, Management and Statistics University of Bologna Academic Year 207/8 NOTES on LINEAR ALGEBRA for the students of Stats and Maths This is a modified version of the notes by Prof Laura
More informationLinear Algebra Primer
Linear Algebra Primer David Doria daviddoria@gmail.com Wednesday 3 rd December, 2008 Contents Why is it called Linear Algebra? 4 2 What is a Matrix? 4 2. Input and Output.....................................
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationMATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam. Topics
MATH 213 Linear Algebra and ODEs Spring 2015 Study Sheet for Midterm Exam This study sheet will not be allowed during the test Books and notes will not be allowed during the test Calculators and cell phones
More informationMath113: Linear Algebra. Beifang Chen
Math3: Linear Algebra Beifang Chen Spring 26 Contents Systems of Linear Equations 3 Systems of Linear Equations 3 Linear Systems 3 2 Geometric Interpretation 3 3 Matrices of Linear Systems 4 4 Elementary
More informationREVIEW FOR EXAM II. The exam covers sections , the part of 3.7 on Markov chains, and
REVIEW FOR EXAM II The exam covers sections 3.4 3.6, the part of 3.7 on Markov chains, and 4.1 4.3. 1. The LU factorization: An n n matrix A has an LU factorization if A = LU, where L is lower triangular
More informationANSWERS. E k E 2 E 1 A = B
MATH 7 Final Exam Spring ANSWERS Essay Questions points Define an Elementary Matrix Display the fundamental matrix multiply equation which summarizes a sequence of swap, combination and multiply operations,
More informationMTH 464: Computational Linear Algebra
MTH 464: Computational Linear Algebra Lecture Outlines Exam 2 Material Prof. M. Beauregard Department of Mathematics & Statistics Stephen F. Austin State University March 2, 2018 Linear Algebra (MTH 464)
More informationMath 321: Linear Algebra
Math 32: Linear Algebra T. Kapitula Department of Mathematics and Statistics University of New Mexico September 8, 24 Textbook: Linear Algebra,by J. Hefferon Email: kapitula@math.unm.edu Prof. Kapitula,
More informationLinear Algebra Practice Problems
Linear Algebra Practice Problems Page of 7 Linear Algebra Practice Problems These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Algebra, 6th ed, by Ron Larson and David Falvo (ISBN3 = 97868783762,
More informationMAT 2037 LINEAR ALGEBRA I web:
MAT 237 LINEAR ALGEBRA I 2625 Dokuz Eylül University, Faculty of Science, Department of Mathematics web: Instructor: Engin Mermut http://kisideuedutr/enginmermut/ HOMEWORK 2 MATRIX ALGEBRA Textbook: Linear
More information3 Matrix Algebra. 3.1 Operations on matrices
3 Matrix Algebra A matrix is a rectangular array of numbers; it is of size m n if it has m rows and n columns. A 1 n matrix is a row vector; an m 1 matrix is a column vector. For example: 1 5 3 5 3 5 8
More informationMatrix & Linear Algebra
Matrix & Linear Algebra Jamie Monogan University of Georgia For more information: http://monogan.myweb.uga.edu/teaching/mm/ Jamie Monogan (UGA) Matrix & Linear Algebra 1 / 84 Vectors Vectors Vector: A
More information1 Matrices and Systems of Linear Equations. a 1n a 2n
March 31, 2013 161 16. Systems of Linear Equations 1 Matrices and Systems of Linear Equations An m n matrix is an array A = (a ij ) of the form a 11 a 21 a m1 a 1n a 2n... a mn where each a ij is a real
More information1. What is the determinant of the following matrix? a 1 a 2 4a 3 2a 2 b 1 b 2 4b 3 2b c 1. = 4, then det
What is the determinant of the following matrix? 3 4 3 4 3 4 4 3 A 0 B 8 C 55 D 0 E 60 If det a a a 3 b b b 3 c c c 3 = 4, then det a a 4a 3 a b b 4b 3 b c c c 3 c = A 8 B 6 C 4 D E 3 Let A be an n n matrix
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationChapter 1: Systems of linear equations and matrices. Section 1.1: Introduction to systems of linear equations
Chapter 1: Systems of linear equations and matrices Section 1.1: Introduction to systems of linear equations Definition: A linear equation in n variables can be expressed in the form a 1 x 1 + a 2 x 2
More informationAPPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF
ELEMENTARY LINEAR ALGEBRA WORKBOOK/FOR USE WITH RON LARSON S TEXTBOOK ELEMENTARY LINEAR ALGEBRA CREATED BY SHANNON MARTIN MYERS APPENDIX: MATHEMATICAL INDUCTION AND OTHER FORMS OF PROOF When you are done
More informationProblem Set (T) If A is an m n matrix, B is an n p matrix and D is a p s matrix, then show
MTH 0: Linear Algebra Department of Mathematics and Statistics Indian Institute of Technology  Kanpur Problem Set Problems marked (T) are for discussions in Tutorial sessions (T) If A is an m n matrix,
More informationLINEAR ALGEBRA SUMMARY SHEET.
LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linearalgebrasummarysheet/ This document is a concise collection of many of the important theorems of linear algebra, organized
More informationLinear Algebra: Lecture Notes. Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway
Linear Algebra: Lecture Notes Dr Rachel Quinlan School of Mathematics, Statistics and Applied Mathematics NUI Galway November 6, 23 Contents Systems of Linear Equations 2 Introduction 2 2 Elementary Row
More informationa s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula
Syllabus for Math 308, Paul Smith Book: KolmanHill Chapter 1. Linear Equations and Matrices 1.1 Systems of Linear Equations Definition of a linear equation and a solution to a linear equations. Meaning
More informationFinal Review Written by Victoria Kala SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015
Final Review Written by Victoria Kala vtkala@mathucsbedu SH 6432u Office Hours R 12:30 1:30pm Last Updated 11/30/2015 Summary This review contains notes on sections 44 47, 51 53, 61, 62, 65 For your final,
More informationSolving a system by backsubstitution, checking consistency of a system (no rows of the form
MATH 520 LEARNING OBJECTIVES SPRING 2017 BROWN UNIVERSITY SAMUEL S. WATSON Week 1 (23 Jan through 27 Jan) Definition of a system of linear equations, definition of a solution of a linear system, elementary
More informationMath Bootcamp An pdimensional vector is p numbers put together. Written as. x 1 x =. x p
Math Bootcamp 2012 1 Review of matrix algebra 1.1 Vectors and rules of operations An pdimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the
More informationIntroduction to Matrices
POLS 704 Introduction to Matrices Introduction to Matrices. The Cast of Characters A matrix is a rectangular array (i.e., a table) of numbers. For example, 2 3 X 4 5 6 (4 3) 7 8 9 0 0 0 Thismatrix,with4rowsand3columns,isoforder
More informationTopic 2 Quiz 2. choice C implies B and B implies C. correctchoice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced rowechelon form of a 3 by 4 matrix can have how many leading one s? choice must have 3 choice may have 1, 2, or 3 correctchoice may have 0, 1, 2, or 3 choice may have 0,
More informationW2 ) = dim(w 1 )+ dim(w 2 ) for any two finite dimensional subspaces W 1, W 2 of V.
MA322 Sathaye Final Preparations Spring 2017 The final MA 322 exams will be given as described in the course web site (following the Registrar s listing. You should check and verify that you do not have
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 2: Matrices, Elimination and Determinant m n matrices The system of m linear equations in n variables x 1, x 2,, x n a 11 x 1 + a 12 x 2 + + a 1n x n = b 1
More informationChapters 5 & 6: Theory Review: Solutions Math 308 F Spring 2015
Chapters 5 & 6: Theory Review: Solutions Math 308 F Spring 205. If A is a 3 3 triangular matrix, explain why det(a) is equal to the product of entries on the diagonal. If A is a lower triangular or diagonal
More informationSolutions to Final Exam
Solutions to Final Exam. Let A be a 3 5 matrix. Let b be a nonzero 5vector. Assume that the nullity of A is. (a) What is the rank of A? 3 (b) Are the rows of A linearly independent? (c) Are the columns
More informationChapter 5 Eigenvalues and Eigenvectors
Chapter 5 Eigenvalues and Eigenvectors Outline 5.1 Eigenvalues and Eigenvectors 5.2 Diagonalization 5.3 Complex Vector Spaces 2 5.1 Eigenvalues and Eigenvectors Eigenvalue and Eigenvector If A is a n n
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationElementary Linear Algebra
Matrices J MUSCAT Elementary Linear Algebra Matrices Definition Dr J Muscat 2002 A matrix is a rectangular array of numbers, arranged in rows and columns a a 2 a 3 a n a 2 a 22 a 23 a 2n A = a m a mn We
More informationhomogeneous 71 hyperplane 10 hyperplane 34 hyperplane 69 identity map 171 identity map 186 identity map 206 identity matrix 110 identity matrix 45
address 12 adjoint matrix 118 alternating 112 alternating 203 angle 159 angle 33 angle 60 area 120 associative 180 augmented matrix 11 axes 5 Axiom of Choice 153 basis 178 basis 210 basis 74 basis test
More informationNOTES FOR LINEAR ALGEBRA 133
NOTES FOR LINEAR ALGEBRA 33 William J Anderson McGill University These are not official notes for Math 33 identical to the notes projected in class They are intended for Anderson s section 4, and are 2
More informationLinear Systems and Matrices
Department of Mathematics The Chinese University of Hong Kong 1 System of m linear equations in n unknowns (linear system) a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.......
More informationMA 265 FINAL EXAM Fall 2012
MA 265 FINAL EXAM Fall 22 NAME: INSTRUCTOR S NAME:. There are a total of 25 problems. You should show work on the exam sheet, and pencil in the correct answer on the scantron. 2. No books, notes, or calculators
More informationMath Camp II. Basic Linear Algebra. Yiqing Xu. Aug 26, 2014 MIT
Math Camp II Basic Linear Algebra Yiqing Xu MIT Aug 26, 2014 1 Solving Systems of Linear Equations 2 Vectors and Vector Spaces 3 Matrices 4 Least Squares Systems of Linear Equations Definition A linear
More information2 Determinants The Determinant of a Matrix Properties of Determinants Cramer s Rule Vector Spaces 17
Contents 1 Matrices and Systems of Equations 2 11 Systems of Linear Equations 2 12 Row Echelon Form 3 13 Matrix Algebra 5 14 Elementary Matrices 8 15 Partitioned Matrices 10 2 Determinants 12 21 The Determinant
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 1 x 2. x n 8 (4) 3 4 2
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS SYSTEMS OF EQUATIONS AND MATRICES Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationMatrix Algebra for Engineers Jeffrey R. Chasnov
Matrix Algebra for Engineers Jeffrey R. Chasnov The Hong Kong University of Science and Technology The Hong Kong University of Science and Technology Department of Mathematics Clear Water Bay, Kowloon
More information2. Linear algebra. matrices and vectors. linear equations. range and nullspace of matrices. function of vectors, gradient and Hessian
FE661  Statistical Methods for Financial Engineering 2. Linear algebra Jitkomut Songsiri matrices and vectors linear equations range and nullspace of matrices function of vectors, gradient and Hessian
More informationChapter 5. Linear Algebra. A linear (algebraic) equation in. unknowns, x 1, x 2,..., x n, is. an equation of the form
Chapter 5. Linear Algebra A linear (algebraic) equation in n unknowns, x 1, x 2,..., x n, is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b where a 1, a 2,..., a n and b are real numbers. 1
More informationk is a product of elementary matrices.
Mathematics, Spring Lecture (Wilson) Final Eam May, ANSWERS Problem (5 points) (a) There are three kinds of elementary row operations and associated elementary matrices. Describe what each kind of operation
More informationKnowledge Discovery and Data Mining 1 (VO) ( )
Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory
More informationExtra Problems for Math 2050 Linear Algebra I
Extra Problems for Math 5 Linear Algebra I Find the vector AB and illustrate with a picture if A = (,) and B = (,4) Find B, given A = (,4) and [ AB = A = (,4) and [ AB = 8 If possible, express x = 7 as
More informationMath 18, Linear Algebra, Lecture C00, Spring 2017 Review and Practice Problems for Final Exam
Math 8, Linear Algebra, Lecture C, Spring 7 Review and Practice Problems for Final Exam. The augmentedmatrix of a linear system has been transformed by row operations into 5 4 8. Determine if the system
More informationMATRICES ARE SIMILAR TO TRIANGULAR MATRICES
MATRICES ARE SIMILAR TO TRIANGULAR MATRICES 1 Complex matrices Recall that the complex numbers are given by a + ib where a and b are real and i is the imaginary unity, ie, i 2 = 1 In what we describe below,
More information7. Dimension and Structure.
7. Dimension and Structure 7.1. Basis and Dimension Bases for Subspaces Example 2 The standard unit vectors e 1, e 2,, e n are linearly independent, for if we write (2) in component form, then we obtain
More informationLinear Algebra Primer
Introduction Linear Algebra Primer Daniel S. Stutts, Ph.D. Original Edition: 2/99 Current Edition: 4//4 This primer was written to provide a brief overview of the main concepts and methods in elementary
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationStudy Guide for Linear Algebra Exam 2
Study Guide for Linear Algebra Exam 2 Term Vector Space Definition A Vector Space is a nonempty set V of objects, on which are defined two operations, called addition and multiplication by scalars (real
More informationChapter 2: Matrices and Linear Systems
Chapter 2: Matrices and Linear Systems Paul Pearson Outline Matrices Linear systems Row operations Inverses Determinants Matrices Definition An m n matrix A = (a ij ) is a rectangular array of real numbers
More informationREVIEW FOR EXAM III SIMILARITY AND DIAGONALIZATION
REVIEW FOR EXAM III The exam covers sections 4.4, the portions of 4. on systems of differential equations and on Markov chains, and..4. SIMILARITY AND DIAGONALIZATION. Two matrices A and B are similar
More informationMTH 102A  Linear Algebra II Semester
MTH 0A  Linear Algebra  056II Semester Arbind Kumar Lal P Field A field F is a set from which we choose our coefficients and scalars Expected properties are ) a+b and a b should be defined in it )
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationSolution to Homework 1
Solution to Homework Sec 2 (a) Yes It is condition (VS 3) (b) No If x, y are both zero vectors Then by condition (VS 3) x = x + y = y (c) No Let e be the zero vector We have e = 2e (d) No It will be false
More informationMatrix Arithmetic. j=1
An m n matrix is an array A = Matrix Arithmetic a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn of real numbers a ij An m n matrix has m rows and n columns a ij is the entry in the ith row and jth column
More informationLINEAR SYSTEMS, MATRICES, AND VECTORS
ELEMENTARY LINEAR ALGEBRA WORKBOOK CREATED BY SHANNON MARTIN MYERS LINEAR SYSTEMS, MATRICES, AND VECTORS Now that I ve been teaching Linear Algebra for a few years, I thought it would be great to integrate
More information