# A Brief Outline of Math 355

Size: px
Start display at page:

## Transcription

1 A Brief Outline of Math 355 Lecture 1 The geometry of linear equations; elimination with matrices A system of m linear equations with n unknowns can be thought of geometrically as m hyperplanes intersecting in R n Some basic questions in this course are: i Are there any points in R n where all the hyperplanes intersect? ii How many such points are there? Geometrically, we saw that in R 2, two lines can intersect either in a single point, everywhere (ie they are the same line), or nowhere (ie they are parallel) We also emphasized the column picture, where we look for a solution as a linear combination of the columns of our matrix The columns are viewed as vectors in R n In order to (attempt to) solve a system of linear equations, we convert the equations to a matrix, then use row operations to reduce a matrix A to an upper triangular matrix U, using which it is easy to backsubstitute and find any solutions Allowable row operations are: i Add a multiple of one row to another ii Exchange rows iii Multiply a row by a nonzero number Lecture 2 Multiplication and inverse matrices; Matrix multiplication (ie AB = C) can be thought of in four ways: 1 One entry at a time: The entry c i,j is the inner product of the ith row of A with the jth column of B 2 A row at a time: The ith row of C is a linear combination of the rows of B, with the coefficients of the linear combination being the ith row of A 3 A column at a time: The ith column of C is a linear combination of the columns of A, with the coefficients of the linear combination being the ith column of B 4 A whole matrix at a time: Multiply a column of A with a row of B to get an m n matrix Add up all such matrices to get AB Given a square, invertible matrix, take the augmented matrix [ A I ], and row reduce A to reduced row echelon form (which will be the identity matrix, I), to get [ I A 1 ] 1

2 Lecture 3 Factorization into A = LU; transposes, permutations, spaces Using the One row at a time idea of multiplication, we could translate row operations into matrix algebra For example, if we were row reducing a 3x3 matrix and wanted to subtract 2 of row 1 from row 2, the picture might look something like: = E 2,1 A = We prefer to reduce a matrix to the form A = LU, since the inverse of the elimination matrices are particularly easy to find (the inverse of a matrix that subtracts 2 of row 2 is one that adds 2 of row 2), and the product E2,1 1 E 1 3,1 E 1 3,2 is just the coefficients from elimination If A = (a i,j ), then the (i, j)th entry of A T, the transpose of A, is (a j,i ) The matrix that permutes the rows of another matrix can be found by performing the permutation on the identity matrix There are n! n n permutation matrices, the inverse of a permutation matrix is its transpose (ie P P T = P P 1 = I), and the product (or transpose) of a permutation matrix is another permutation matrix A vector space is a collection V of objects (which are called vectors), which can be added or multiplied by a (real) number (and the result will still be in V ) A subspace of a vector space V is a subset of V which is still a vector space For example, the column space of an m n matrix is a subspace of R m Lecture 4 R n ; column space and null space; solving Ax = 0: pivot variables, special solutions Our prime example of a vector space is R n, that is, we take our vectors to be n-tuples of real numbers, and perform addition and scalar multiplication componentwise The column space of a matrix A is the set of all linear combinations of the columns of a matrix Equivalently, it is the set of all b so that Ax = b has a solution We denote the column space by C(A) The null space of a matrix A is the set of all x so that Ax = 0 We denote the null space by N(A) To find the null space of a matrix A: 2

3 i Use Gauss-Jordan elimination to convert A to reduced row echelon form, R You will have r pivot variables and n r free variables ii Set the first free variable equal to 1 and the rest equal to 0, then solve for the pivot variables This is the first special solution iii Repeat the previous step with each of the other free variables to find n r linearly independent special solutions iv These special solutions form a basis for N(A) (so any linear combination of the special solutions is in the null space) Lecture 5 Solving Ax = b: row reduced form R; independence, span, basis and dimension Algorithm for complete solution to Ax = b: i Use row operations to change A to R ii Set free variables to zero, solve for pivot variables to find x particular iii Find the nullspace of A, N(A) iv Complete solution to Ax = b is x = x p + x n, where x n is any vector in N(A) So a complete solution to such a problem would consist of finding some x p (remember there are infinitely many), and N(A) (plus writing x = x p + x n ) Given an m n matrix A with rank r, there are three special cases: i r = n < m: Then N(A) = {0}, and Ax = b has either 0 or 1 solution ii r = m < n: Then dim(c(a)) = r = m, so C(A) = R m, and so Ax = b always has a solution Also, dim(n(a)) = n r > 0, so there are in fact many solutions iii r = m = n: Then N(A) = {0} and Ax = b always has a solution Thus there is always a unique solution to Ax = b A set of vectors v 1, v 2,, v n are linearly independent if a 1 v 1 + a 2 v a n v n = 0 means that a 1 = a 2 = = a n = 0 The algorithm for checking whether vectors are independent is to create a matrix A with the vectors as columns If N(A) = 0, then the vectors are independent Otherwise, they are dependent 3

4 The span of a set of vectors is all the linear combinations of those vectors We say that v 1, v 2,, v n span a vector space V if V = span{v 1, v 2,, v n } For example, the span of the columns of a matrix is the column space A set of vectors is a basis for a vector space V if i The vectors are linearly independent, and ii The vectors span V The dimension of a vector space V is the number of vectors in any basis for V (recall, we showed that any basis for V has the same number of vectors) We also showed that if dim(v ) = n, then any n linearly independent vectors in V will be a basis for V Lecture 6 The four fundamental subspaces; matrix spaces, polynomial spaces We took an m n matrix A, and looked at the column space(c(a)), the null space (N(A)), the row space (C(A T )), and the left null space (N(A T )) The natural questions to ask when looking at subspaces are: 1 What is a basis? 2 What is the dimension? We answer these questions here: Suppose we have a matrix A Then if we take the augmented matrix [ A I ] and use row operations to reduce A to reduced row echelon form, R, then we call the matrix on the right E, for elimination matrix (note the matrix E s relationship to the elimination matrices from chapter 1) That is, we use row reduction to go from: [ A I ] [ R E ] Now we can easily read off the rank, r of the matrix A, by counting the pivot variables in R, as well as calculate: Dimension of C(A): Is just the rank, r Basis for C(A): Is the r columns of A that correspond to the pivot columns of R Dimension of C(A T ): Is also the rank, r Basis for C(A T ): These are the first r ows of R (since row operations do not change the row space, and the first r rows are the pivot rows) Dimension of N(A): This is the number of free variables, which is n r 4

5 Basis for N(A): We find n r solutions to the system of equations Rx = 0, by setting one free variable equal to 1 at a time, while leaving the rest equal to zero, and solving Dimension of N(A T ): Since this is just the null space of A T, which has r pivots and m r free variables, this must have dimension m r Basis for N(A T ): Take the bottom m r rows of E The space M m n of m n matrices can also be considered a vector space, even though matrices are not traditionally thought of as vectors We also inspected the subspaces of upper triangular, symmetric and diagonal matrices Be able to find bases for these spaces As an example, a basis for M 3 3 is , ,, The space P n of polynomials of degree n is also a vector space of dimension n + 1, with basis {1, x, x 2,, x n } Lecture 7 Graphs, networks, incidence matrices A graph consists of nodes and edges If these were more serious notes, there d be an example drawn An incidence matrix for an oriented graph with m edges and n nodes will be an m n matrix, with the entries 1 if edge i leaves node j a i,j = 1 if edge i enters node j 0 otherwise Each of the four fundamental subspaces has a physical interpretation, starting with interpreting the vector x as the potential at each node: The column space The vector e = Ax represents the possible potential differences The null space This is the stationary solution- when there is no potential difference The left null space The set of y so that Ay = 0 are those currents which satisfy Kirchoff s circuit law, which says that the net flow of current at any node must be 0 The row space The corresponding pivot rows will create a maximum tree in the graph (ie a subgraph that has no loops, but contains every node) 5

6 An incidence matrix has another interesting interpretation: the dimension of N(A T ) is the number of loops in the graph, while the rank is the number of nodes, minus 1 Hence, dim(n(a T )) = m r # loops = # edges # of nodes 1 This is Euler s formula Lecture 8 Orthogonal vectors and subspaces; projections onto subspaces Two vectors x and y are orthogonal if x T y = 0 Two subspaces S and T are orthogonal if s T t = 0 for every vector s S and t T Two subspaces S and T of R n are orthogonal complements if i S and T are orthogonal ii dims + dimt = n The row space and null space of an m n matrix are orthogonal complements in R n The column space and the left null space of an m n matrix are orthogonal complements in R m To project a vector b onto the subspace generated by a, we use the projection matrix P, given by P = aat a T a Then the projection of b is just P b We define that a projection matrix is any matrix so that i P T = P, and ii P 2 = P Lecture 9 Projection matrices and least squares; orthogonal matrices and Gram-Schmidt We use projections to solve least squares problems That is, in the event that there is no x so that Ax = b, we find an ˆx so that Aˆx = b and x ˆx 2 is as small as possible We solve least squares using the projection matrix in the sense that P b = Aˆx P = A(A T A) 1 A T, 6

7 In practice, to solve the least squares problem, you solve the equation A T Aˆx = A T b This will have a solution if and only if A T A is invertible, which is true whenever A has independent columns A set of vectors q 1, q 2,, q n is orthonormal if { q T 1 if i = j i q j = 0 if i j Any matrix (rectangular or square) Q with orthonormal columns has the property Q T Q = I If Q is also square, then Q T = Q 1 If we have a least squares problem with an orthogonal matrix (ie one with orthonormal columns), then the projection equation simplifies to ˆx = Q T b, so in particular, the i th coordinate of ˆx is ˆx i = q T i b The Gram-Schmidt process takes a set of vectors a, b, c, z (ok, I don t mean precisely 26 vectors, but I don t want to involve subscripts either so bear with me), and converts them into orthogonal vectors A, B, C,, Z, and then into orthonormal vectors q 1, q 2, q 3,, q n, so that all of the different sets of vectors have the same span Here is the algorithm: 1 We define the orthogonal vectors recursively: A = a, B = b AT b A T A, C = c AT c A T A BT c B T B, Z = z AT z A T A BT z B T B YT z Y T Y 7

8 2 We normalize the vectors: q 1 = q 2 = q 3 = q n = A A B B C C Z Z (1) Lecture 10 Properties of determinants; determinant formulas and cofactors We deduced that three properties of the determinant completely determine the determinant We used these three properties to prove that seven more properties hold, and then used this to deduce formulas for the determinant The determinant is a function that eats square (real valued) matrices and gives a (real) number The three defining properties of the determinant are: 1 deti = 1 2 Transposing two rows of a matrix changes the sign of the determinant 3 The determinant is linear in each row This means: a Multiplying a row by a number multiplies the determinant by the same number For example, if A = r 1 r 2 r i r n, and A = r 1 r 2 tr i r n, then deta = t deta b Adding a vector to a row of a matrix is additive (this doesn t seem like the right word to use, but I don t think a correct and simple word exists) For 8

9 example, if r 1 r 2 A r = r i, A s = r n and A = then deta = deta r + deta s r 1 r 2 r i + s i r n r 1 r 2 s i r n,, We then used the previous three properties to deduce seven more properties that the determinant must satisfy: 4 If two rows of A are equal, then deta = 0 5 Subtracting k (row i) from row j does not change the determinant 6 If A has a row of zeros, then deta = 0 7 The determinant of an upper triangular matrix is the product of the pivots: d 1 u 1,2 u 1,n 0 d 2 u 2,n detu = = d 1 d 2 d n 0 0 d n 8 deta = 0 if and only if A is singular 9 det(ab) = (deta)(detb) 10 deta = deta T We then used the above 10 properties to determine 3 formulas for the determinant of a matrix: Long formula with n! terms: By expanding a matrix using property 3b, and eliminating those with rows of 0 using property 6, we got deta = ±a 1,α a 2,β a 3,γ a n,ω, n! permutations of 1,n 9

10 where {α, β, γ,, ω} is some permutation of {1, 2, 3,, n}, and the sign is determined by whether this is an odd or even permutation Cofactor expansion: The cofactor of a i,j, denoted c i,j, is c i,j = ( 1) i+j det Then we concluded that [ (n 1) (n 1) matrix with row i, col j removed deta = a 1,1 c 1,1 + a a,2 c 1,2 + + a 1,n c 1,n, and referred to this as the cofactor expansion along row 1 A similar formula holds expanding along any row or column Row reduction: Using properties 5 and 7, we concluded that row reducing a matrix to A = LU, then deta = detu = the product of the pivots This is the most computationally efficient method in general, though cofactors are also very useful in computing by hand Lecture 11 Applications of the determinant: Cramer s rule, inverse matrices, and volume; eigenvalues and eigenvectors We defined C to be the cofactor matrix of A: that is, c i,j is the cofactor associated with a i,j This allowed us to write A 1 = 1 deta C (note this equation only holds if A is invertible) Cramer s Rule gives us an explicit way of solving for each coordinate of Ax = b In particular, x 1 = detb 1 deta x 2 = detb 2 deta x n = detb n deta, where B i is the matrix A with the i th column replaced by the vector b We also saw that the volume of an n-dimensional parallelepiped with edges a 1, a 2,, a n is the absolute value of the determinant of the matrix A with columns a 1, a 2,, a n ] 10

11 An eigenvalue of a matrix A is a number λ so that there exists a vector x (called the eigenvector) with Ax = λx To find the eigenvalues of A, we solve the characteristic equation det[a λi] = 0, which will be an n th degree polynomial (and so will have n not-necessarily-distinct, not-necessarily real roots) To find the eigenvectors, we take the eigenvalues λ 1,, λ n, and let x i be a vector in the nullspace of A λ i I (this is a little imprecise, since if two eigenvalues are the same, the nullspace of A λi may contain more than one linearly independent vector) If we have n independent eigenvectors, and put them as columns of the matrix S, then S 1 AS = Λ and A = SΛS 1, where Λ is a diagonal matrix with the eigenvalues along the diagonal: λ λ 2 0 Λ = 0 0 λ n You can remember this equation since Ax i = x i λ i corresponds to multiplying A on the right by the column x i, giving AS = SΛ Note that if a matrix can be diagonalized, then A k = SΛ k S 1, where Λ k is easily computed as λ k Λ k 0 λ k 2 0 = 0 0 λ k n A matrix is diagonalizable if and only if it has n independent eigenvectors If each of the eigenvalues are different, then the matrix is sure to be diagonalizable However, if a matrix has repeated eigenvalues, then it may or may not be diagonalizable 11

12 Solved the equation: u k+1 = Au k, given the initial vector u 0, by noting that u k = A k u 0 To actually compute u k : i Find eigenvalues λ 1,, λ n and eigenvectors x 1,, x n of A, ii Write u 0 = c 1 x 1 + c 2 x c n x n = Sc, where S is the eigenvector matrix, and c = [c 1,, c n ] T is the solution vector to Sc = u 0 iii Then u k = Λ k Sc Lecture 12 Diagonalization and powers of A; differential equations and e At Solved linear equations of the form du 1 dt du 2 dt du n dt = a 1,1 u 1 + a 1,2 u a 1,n u n = a 2,1 u 1 + a 2,2 u a 2,n u n = a n,1 u 1 + a n,2 u a n,n u n, which we wrote in the decidedly more compact form du dt = Au We typically are also given an initial condition u(0) To solve: i Find eigenvalues λ 1,, λ n and eigenvectors x 1,, x n of A, ii Solution is u(t) = c 1 e λ1t x 1 + c 2 e λ2t x c n e λnt x n, where c = [c 1,, c n ] T is found by noting that u(0) = Sc This can also be written u(t) = Se Λt S 1 u(0) We noted that the exponential of a matrix is defined by: e At = I + At + (At)2 2! + (At)3 3! + = Se Λt S 1, (2) with the second equality holding only if A is diagonalizable 12

13 We also saw that e Λt = e λ1t e λ2t e λnt You can change a single 2 nd order equation into a system of 1 st order equations by rewriting y + by + ky = 0 as ( y u = y ) (, so u y = y ) = ( b k 1 0 ) ( y y ) This can also be used to reduce n th order differential equations to a system of n first order equations Lecture 13 Markov matrices, Fourier series A Markov matrix is one where i All entries 0 ii The entries in each column add to 1 If A is a Markov matrix, then λ = 1 is an eigenvalue, and λ i 1 for all other eigenvalues Hence the steady state will be some multiple of the eigenvector x 1 corresponding to λ 1 = 1 Given an orthonormal basis q 1, q 2,, q n, we can write any v as v = x 1 q 1 + x 2 q c n q n Since the q i s are orthonormal, multiplying the equation on the left by q T i leaves us with q T i v = x i The Fourier series for a function f(x) is the expansion f(x) = a 0 + a 1 cos x + b 1 sin x + a 2 cos 2x + b 2 sin 2x + We define the inner product for these functions as f T g = 2π 0 f(x)g(x)dx, 13

14 Lecture 14 Symmetric Matrices and observe that 1, cos x, sin x, cos 2x, sin 2x, is an orthogonal basis (though each one has norm π, so it is easy to make it orthonormal) Hence to find b 2 (for example), we use the above and observe b 2 = 1 π 2π 0 f(x) sin 2x dx If you have a symmetric matrix (that is, A = A T, or when a complex matrix, A = ĀT ), then 1 The eigenvalues of A are real, and 2 The eigenvectors of A can be chosen to be orthogonal Then a symmetric matrix A can be factored as A = QΛQ T (compare to the usual case A = SΛS 1 ) This is called the spectral theorem Multiplying out the factorization above, we get where each A = λ 1 q 1 q T 1 + λ 2 q 2 q T λ n q n q T n, q i q T i = q iq T i q T i q i is an orthogonal projection matrix So every symmetric matrix is a linear combination of orthogonal projection matrices For a symmetric matrix, the number of positive pivots is the same as the number of positive eigenvalues A positive definite matrix is a symmetric matrix where all eigenvalues are positive (which is the same as all the pivots being positive) Lecture 15 Complex matrices and the Fast Fourier Transform A complex number z can be written in three ways: i z = x + iy It can be viewed on the complex plane as the point (x, y), making the obvious identification with R 2 ii z = r(cos θ + i sin θ) In this case, r is called the modulus (fancy word for length ) of z, and θ is called the argument It can be viewed on the complex plane as the endpoint of the vector leaving from the origin with lengh r and angle θ 14

15 iii z = re iθ See above for the terminology This form has the same geometric interpretation as ii, but is more widely used For example, 2i = 2e πi 2 The complex conjugate of a complex number z is found by switching the sign on the imaginary part of z, or graphically by reflecting z over the real axis, and is denoted by z: If z = x + iy = re iθ, then z = x iy = re iθ A number z is real if and only if z = z The length of a complex number z is (z z) 1 2 = (x + iy)(x iy) = x2 + y 2 = r Given a complex vector z C n, z 1 z 2 z =, z n we noticed that the length was given by z T z, and so defined the Hermitian as the transpose of the conjugate: For vectors, z H := z T For complex matrices, A H := ĀT We use the Hermitian to translate words we used for real matrices and vectors into words for complex matrices and vectors: Def for R-valued Def for C-valued Length of x x T x x H x Inner product x T y x H y A symmetric A = A T { A = A H { q 1,, q n q T i q j = 0 if i j q H i 1 otherwise 0 if i j j = 1 otherwise orthonormal Notice that the only difference is that the transpose is always exchanged for a Hermitian, and that when dealing with real vectors/matrices, each definition is the same The n th Fourier matrix, F n is defined as ω ω 2 ω n 1 F n = 1 ω 2 ω 4 ω 2(n 1), 1 ω n 1 ω 2(n 1) ω (n 1)(n 1) 15

16 where ω is the n th root of unity, that is, ω is a solution of x n 1 = 0 More specifically, ω = e 2πi n The columns of F n are orthogonal (so F H n F n = I), and can be multiplied very quickly Lecture 16 Positive definite matrices and minima; Similar matrices and Jordan form We looked at four equivalent definitions for an n n matrix A being positive definite: i λ 1 > 0, λ 2 > 0,, λ n > 0 ii Each of the n leading subdeterminants are strictly positive The m th leading subdeterminant is the determinant of the m m matrix in the top left corner of A iii Each of the pivots of A are strictly positive (Careful: this does not mean that the diagonal of A is positive It means that if A = LU, then the elements on the diagonal of U are positive!) iv x T Ax > 0 for all x We define positive semidefinite by replacing all the incidences of the words strictly positive above by positive or zero The terms negative definite and negative semidefinite are defined the same, just replacing positive by negative in the definition The function produced by x T Ax is called a quadratic form When A is 2 2, this corresponds to a conic section If A is positive definite, then x T Ax is a paraboloid More generally, let f : R n R (think: f(x 1, x 2,, x n ) = y) then if f(a) = 0 (where f = ( f x 1, f x 2,, f x n )) we have: 2 f 2 f x 2 1 x 1 x 2 2 f 2 f 2 f f(a) is a minimum if f x (a) = 2 x 1 x f x n x 2 2 f x n x 1 x 1 x n 2 f x 2 x n 2 f x 2 n is positive definite (where each second derivative is evaluated at a) Compare this to calculus where a is a minimum if f (a) = 0 and f (a) > 0 Positive definite matrices act like positive numbers: If A, B are positive definite matrices, then so are A 1 and A + B Also, A T A is positive definite for any m n matrix A with rank n (since x T A T Ax = (Ax) T (Ax) = Ax 2 > 0) 16

17 Two n n matrices are similar if there is an invertible matrix M so that B = M 1 AM An example to remember is that every diagonalizable matrix is similar to a diagonal matrix (A = SΛS 1 ) Similar matrices have the same eigenvalues, and represent the same linear transformations with different coordinates We found a good representative for each family of matrices (that is to say, a family of matrices is the set of all matrices you can get by conjugating the matrix by an invertible matrix ie the set of matrices similar to eachother), which we called the Jordan canonical form A Jordan block is the matrix λ λ 1 0 J λ = 0 0 λ λ Every matrix A is similar to a Jordan canonical matrix, which looks like J λ J λ2 0 J = 0 0 J λn Lecture 17 Singular Value Decomposition; Linear transformations and their matrices, coordinates The singular value decomposition works for all matrices, and decomposes the m n matrix A into A = UΣV T, where U is an m m orthogonal matrix, Σ is an m n diagonal matrix with all entries 0, and V is an n n orthogonal matrix The columns of U are the eigenvectors of A T A (which, recall, is positive indefinite), and the diagonal entries of Σ are the square roots of the associated eigenvalues The columns of V are the eigenvectors of AA T, and again the diagonal entries of Σ are square roots of the eigenvalues We can also look at the columns v 1,, v n of V and the columns u 1,, u m of U in the following way: Let r be the rank of A 17

18 Lecture 18 Change of basis v 1,, v r are an orthonormal basis for C(A), u 1,, u r are an orthonormal basis for C(A T ), v r+1,, v n are an orthonormal basis for N(A T ), u r+1,, u m are an orthonormal basis for N(A) A linear transformation is a function T : R n R m so that i T (u + v) = T (u) + T (v), and ii T (cv) = ct (v) Given coordinates, that is, a basis for R n and R m, every linear transformation T is uniquely associated with a matrix A Translating a linear transformation into a matrix: 1 You will be given a linear transformation T : R n R m, as well as a basis v 1,, v n of R n and a basis u 1,, u m of R m (in practice, you may decide the bases) 2 Evaluate the basis elements of R n, and write them in the coordinates of R m : T (v 1 ) = a 1,1 w 1 + a 2,1 w a m,1 w m T (v 2 ) = a 1,2 w 1 + a 2,2 w a m,2 w m T (v n ) = a 1,n w 1 + a 2,n w a m,n w m 3 Now A = (a i,j ) will be the matrix representation of the linear transformation in the given basis Two matrices represent the same linear transformation in different coordinates precisely when they are similar, which is one good reason to use eigenvectors as coordinates (so that the linear transformation matrix is diagonal) A natural question is : if a linear transformation T : R n R m has matrix A with respect to the basis v 1,, v n, and matrix B with respect to the basis u 1,, u m then what is the relationship between A and B? Denote R n by V or U, depending on the basis used Then we want to find a matrix M so that A V V M M B U U commutes That is to say, B = M 1 AM But M may be interpreted as a linear transformation, with Mv i = v i = a 1,i u 1 + a 2,i u a n,i u n, and we use the above to find the matrix M 18

### Math 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

### Linear 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.....................................

### Conceptual 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.

### [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

### Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Math 520 Exam 2 Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008 Exam 2 will be held on Tuesday, April 8, 7-8pm in 117 MacMillan What will be covered The exam will cover material from the lectures

### MATH 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

### 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 +...

### Problem 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,

### IMPORTANT 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

### (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? Solution: dim N(A) 1, since rank(a) 3. Ax =

. (5 points) (a) If A is a 3 by 4 matrix, what does this tell us about its nullspace? dim N(A), since rank(a) 3. (b) If we also know that Ax = has no solution, what do we know about the rank of A? C(A)

### Math Linear Algebra Final Exam Review Sheet

Math 15-1 Linear Algebra Final Exam Review Sheet Vector Operations Vector addition is a component-wise operation. Two vectors v and w may be added together as long as they contain the same number n of

### IMPORTANT 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

### 33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM

33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the

### Review 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

### 1. 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

### MAT Linear Algebra Collection of sample exams

MAT 342 - Linear Algebra Collection of sample exams A-x. (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

### HOMEWORK 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

### 1 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

### Math 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

### Linear 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

### Linear 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

### Math113: 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

### Eigenvalues and Eigenvectors

/88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix

### LINEAR ALGEBRA SUMMARY SHEET.

LINEAR ALGEBRA SUMMARY SHEET RADON ROSBOROUGH https://intuitiveexplanationscom/linear-algebra-summary-sheet/ This document is a concise collection of many of the important theorems of linear algebra, organized

### MATH 583A REVIEW SESSION #1

MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),

### Linear Algebra Highlights

Linear Algebra Highlights Chapter 1 A linear equation in n variables is of the form a 1 x 1 + a 2 x 2 + + a n x n. We can have m equations in n variables, a system of linear equations, which we want to

### ANSWERS. 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,

### 2. 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

### Final 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 5.1-5.4, 6.1-6.2 and 7.1-7.3 from chapters 5,6 and 7. This is essentially all material covered this term. Watch

### MATH 1120 (LINEAR ALGEBRA 1), FINAL EXAM FALL 2011 SOLUTIONS TO PRACTICE VERSION

MATH (LINEAR ALGEBRA ) FINAL EXAM FALL SOLUTIONS TO PRACTICE VERSION Problem (a) For each matrix below (i) find a basis for its column space (ii) find a basis for its row space (iii) determine whether

### 18.06SC Final Exam Solutions

18.06SC Final Exam Solutions 1 (4+7=11 pts.) Suppose A is 3 by 4, and Ax = 0 has exactly 2 special solutions: 1 2 x 1 = 1 and x 2 = 1 1 0 0 1 (a) Remembering that A is 3 by 4, find its row reduced echelon

### Linear Algebra Lecture Notes

Linear Algebra Lecture Notes jongman@gmail.com January 19, 2015 This lecture note summarizes my takeaways from taking Gilbert Strang s Linear Algebra course online. 1 Solving Linear Systems 1.1 Interpreting

### Math 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

### 1. 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 +

### ANSWERS (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 Cauchy-Schwartz inequality

### Linear algebra and applications to graphs Part 1

Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces

### MIT Final Exam Solutions, Spring 2017

MIT 8.6 Final Exam Solutions, Spring 7 Problem : For some real matrix A, the following vectors form a basis for its column space and null space: C(A) = span,, N(A) = span,,. (a) What is the size m n of

### There are six more problems on the next two pages

Math 435 bg & bu: Topics in linear algebra Summer 25 Final exam Wed., 8/3/5. Justify all your work to receive full credit. Name:. Let A 3 2 5 Find a permutation matrix P, a lower triangular matrix L with

### Math 21b. Review for Final Exam

Math 21b. Review for Final Exam Thomas W. Judson Spring 2003 General Information The exam is on Thursday, May 15 from 2:15 am to 5:15 pm in Jefferson 250. Please check with the registrar if you have a

### Calculating determinants for larger matrices

Day 26 Calculating determinants for larger matrices We now proceed to define det A for n n matrices A As before, we are looking for a function of A that satisfies the product formula det(ab) = det A det

### 4. Determinants.

4. Determinants 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 2 2 determinant 4.1. Determinants; Cofactor Expansion Determinants of 2 2 and 3 3 Matrices 3 3 determinant 4.1.

### MATH 235. Final ANSWERS May 5, 2015

MATH 235 Final ANSWERS May 5, 25. ( points) Fix positive integers m, n and consider the vector space V of all m n matrices with entries in the real numbers R. (a) Find the dimension of V and prove your

### 18.06 Problem Set 8 - Solutions Due Wednesday, 14 November 2007 at 4 pm in

806 Problem Set 8 - Solutions Due Wednesday, 4 November 2007 at 4 pm in 2-06 08 03 Problem : 205+5+5+5 Consider the matrix A 02 07 a Check that A is a positive Markov matrix, and find its steady state

### Glossary 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

### MATH 31 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL

MATH 3 - ADDITIONAL PRACTICE PROBLEMS FOR FINAL MAIN TOPICS FOR THE FINAL EXAM:. Vectors. Dot product. Cross product. Geometric applications. 2. Row reduction. Null space, column space, row space, left

### ELEMENTARY 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

### Solving a system by back-substitution, 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

### MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003

MATH 23a, FALL 2002 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Solutions to Final Exam (in-class portion) January 22, 2003 1. True or False (28 points, 2 each) T or F If V is a vector space

### MTH Linear Algebra. Study Guide. Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education

MTH 3 Linear Algebra Study Guide Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education June 3, ii Contents Table of Contents iii Matrix Algebra. Real Life

### MATRIX 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

### Linear Algebra, Summer 2011, pt. 2

Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................

### Math 302 Outcome Statements Winter 2013

Math 302 Outcome Statements Winter 2013 1 Rectangular Space Coordinates; Vectors in the Three-Dimensional Space (a) Cartesian coordinates of a point (b) sphere (c) symmetry about a point, a line, and a

### Chapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors

Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there

### Math 307 Learning Goals

Math 307 Learning Goals May 14, 2018 Chapter 1 Linear Equations 1.1 Solving Linear Equations Write a system of linear equations using matrix notation. Use Gaussian elimination to bring a system of linear

### Equality: 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 m-by-n 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

### Online 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

### Chapter 3 Transformations

Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases

### Study 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

### MATH 310, REVIEW SHEET 2

MATH 310, REVIEW SHEET 2 These notes are a very short summary of the key topics in the book (and follow the book pretty closely). You should be familiar with everything on here, but it s not comprehensive,

### Solutions to Final Practice Problems Written by Victoria Kala Last updated 12/5/2015

Solutions to Final Practice Problems Written by Victoria Kala vtkala@math.ucsb.edu Last updated /5/05 Answers This page contains answers only. See the following pages for detailed solutions. (. (a x. See

### Ir 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

### Chapter 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

### Math Bootcamp An p-dimensional 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 p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the

### NOTES 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

### DS-GA 1002 Lecture notes 0 Fall Linear Algebra. These notes provide a review of basic concepts in linear algebra.

DS-GA 1002 Lecture notes 0 Fall 2016 Linear Algebra These notes provide a review of basic concepts in linear algebra. 1 Vector spaces You are no doubt familiar with vectors in R 2 or R 3, i.e. [ ] 1.1

### Lecture 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

### SYLLABUS. 1 Linear maps and matrices

Dr. K. Bellová Mathematics 2 (10-PHY-BIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V

### The 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

### Linear Algebra. Min Yan

Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................

### 1 Last time: determinants

1 Last time: determinants Let n be a positive integer If A is an n n matrix, then its determinant is the number det A = Π(X, A)( 1) inv(x) X S n where S n is the set of n n permutation matrices Π(X, A)

### SUMMARY 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

### (b) If a multiple of one row of A is added to another row to produce B then det(b) =det(a).

.(5pts) Let B = 5 5. Compute det(b). (a) (b) (c) 6 (d) (e) 6.(5pts) Determine which statement is not always true for n n matrices A and B. (a) If two rows of A are interchanged to produce B, then det(b)

### MATH 369 Linear Algebra

Assignment # Problem # A father and his two sons are together 00 years old. The father is twice as old as his older son and 30 years older than his younger son. How old is each person? Problem # 2 Determine

### LINEAR ALGEBRA QUESTION BANK

LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,

### Daily 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

### BASIC ALGORITHMS IN LINEAR ALGEBRA. Matrices and Applications of Gaussian Elimination. A 2 x. A T m x. A 1 x A T 1. A m x

BASIC ALGORITHMS IN LINEAR ALGEBRA STEVEN DALE CUTKOSKY Matrices and Applications of Gaussian Elimination Systems of Equations Suppose that A is an n n matrix with coefficents in a field F, and x = (x,,

### A Review of Linear Algebra

A Review of Linear Algebra Mohammad Emtiyaz Khan CS,UBC A Review of Linear Algebra p.1/13 Basics Column vector x R n, Row vector x T, Matrix A R m n. Matrix Multiplication, (m n)(n k) m k, AB BA. Transpose

### Linear Algebra: Matrix Eigenvalue Problems

CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given

### MATH 315 Linear Algebra Homework #1 Assigned: August 20, 2018

Homework #1 Assigned: August 20, 2018 Review the following subjects involving systems of equations and matrices from Calculus II. Linear systems of equations Converting systems to matrix form Pivot entry

### Linear 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:

### MTH 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)

### Math 307 Learning Goals. March 23, 2010

Math 307 Learning Goals March 23, 2010 Course Description The course presents core concepts of linear algebra by focusing on applications in Science and Engineering. Examples of applications from recent

### Math 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.

Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,

### 18.06 Professor Johnson Quiz 1 October 3, 2007

18.6 Professor Johnson Quiz 1 October 3, 7 SOLUTIONS 1 3 pts.) A given circuit network directed graph) which has an m n incidence matrix A rows = edges, columns = nodes) and a conductance matrix C [diagonal

### Basic Elements of Linear Algebra

A Basic Review of Linear Algebra Nick West nickwest@stanfordedu September 16, 2010 Part I Basic Elements of Linear Algebra Although the subject of linear algebra is much broader than just vectors and matrices,

### Mathematical Methods wk 2: Linear Operators

John Magorrian, magog@thphysoxacuk These are work-in-progress notes for the second-year course on mathematical methods The most up-to-date version is available from http://www-thphysphysicsoxacuk/people/johnmagorrian/mm

### Eigenvalues and Eigenvectors

Sec. 6.1 Eigenvalues and Eigenvectors Linear transformations L : V V that go from a vector space to itself are often called linear operators. Many linear operators can be understood geometrically by identifying

### Math 215 HW #9 Solutions

Math 5 HW #9 Solutions. Problem 4.4.. If A is a 5 by 5 matrix with all a ij, then det A. Volumes or the big formula or pivots should give some upper bound on the determinant. Answer: Let v i be the ith

### homogeneous 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

### Reduction to the associated homogeneous system via a particular solution

June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one

### MATH 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

### OHSx 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)

### LINEAR ALGEBRA REVIEW

LINEAR ALGEBRA REVIEW SPENCER BECKER-KAHN 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

### Linear Algebra Review

Chapter 1 Linear Algebra Review It is assumed that you have had a course in linear algebra, and are familiar with matrix multiplication, eigenvectors, etc. I will review some of these terms here, but quite

### Notes on Linear Algebra

1 Notes on Linear Algebra Jean Walrand August 2005 I INTRODUCTION Linear Algebra is the theory of linear transformations Applications abound in estimation control and Markov chains You should be familiar

### Third Midterm Exam Name: Practice Problems November 11, Find a basis for the subspace spanned by the following vectors.

Math 7 Treibergs Third Midterm Exam Name: Practice Problems November, Find a basis for the subspace spanned by the following vectors,,, We put the vectors in as columns Then row reduce and choose the pivot

### Maths for Signals and Systems Linear Algebra in Engineering

Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE

### 5. Orthogonal matrices

L Vandenberghe EE133A (Spring 2017) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal