SYLLABUS. 1 Linear maps and matrices
|
|
- Anastasia Copeland
- 5 years ago
- Views:
Transcription
1 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 ) is a vector space. Matrix of linear map. Set of matrices M m,n (F ). Thm 1.1.2: Given bases of n- dimensional U and m-dimensional V, the map T A T is a bijection of the sets L(U, V ) and M m,n (F ). Matrix of sum of linear maps and sum of matrices. Matrix of map multiplied by scalar and multiplication of matrices by scalars. Prop 1.1.3: 1) M m,n (F ) is a vector space; 2) Given bases of n-dimensional U and m-dimensional V, the map T A T is an isomorphism between L(U, V ) and M m,n (F ). Matrix of composition of linear maps and product of matrices. Prop 1.1.4: 1) A S T = A S A S ; 2+3) Product of matrices is an associative operation but generally non-commutative. Row and column vectors. Identity matrix, inverse matrix, invertible matrix. Inverse of a map. Review: map f : X Y is invertible f is a bijection; (f 1 ) 1 = f. Prop 1.1.5: If T : U V is a linear bijeciton, then T 1 : V U is linear. Thm 1.1.6: Map is invertible if and only if its matrix (in some bases) is invertible; moreover, (A T ) 1 = A T 1. Systems of linear equations: Consistent and inconsistent systems. Equivalent systems. Elementary operations with systems. Thm 1.1.7: Elementary operations keep the systems equivalent. Matrix and augmented matrix of the system. Elementary row operations with matrices. Row echelon form and reduced row echelon form. Thm [Gaussian elimination]: Every matrix can be transformed to row echelon form, and hence to reduced row echelon form, by elementary operations. Algorithm for solving a system of linear equations. Homogeneous system. The solutions set to a homogeneous system as a vector space. Example: equation of a plane in space, corresponding homogeneous equation. Rank of a linear map. Prop 1.1.9: For a linear map T : U V, the inverse T 1 exists if and only if dim U = dim V = rank T. Rank of matrix as maximal number of linearly independent columns. Thm : Rank of a linear map equals to rank of its matrix (in some bases), i.e., rank T = rank A T. Corollary : A matrix A M n (F ) is invertible if and only if its rank is n. Thm : Rank of matrix is equal to the maximal number of linearly independent rows. Rank is preserved under elementary row and column transformations. Computation of the rank of a matrix using Gaussian elimination. Finding a maximal subset of linearly independent vectors among a given finite set of vectors. Computation of inverse matrix using Gaussian elimination. Theorem : For all A, B M n (F ), rank(ab) min (rank(a), rank(b)). Corollary : A M n (F ) is invertible if and only if B : AB = I n (then
2 B = A 1 ). Two matrices A, B M n (F ) are both invertible if and only if their product AB is invertible. Example of rank(ab) rank(ba) (HW). Prop : Dimension of solution set to homogeneous system of equation as the number of unknowns minus the rank of the system matrix. Fundamental system of solutions. Permutations. Row notation. Cycle decomposition. Composition of permutations. Transposition. Thm : (1) each permutation can be written as a composition of transpositions, (i 1... i k ) = (i 1 i k )(i 1 i k 1 )... (i 1 i 3 )(i 1 i 2 ); (2) for a given permutation, the number of transpositions in any such decomposition is always even or always odd. Signature (or sign) of a permutation. Natural properties of a volume function. Multilinear function. Alternating function. Proposition : If f : V n F is multilinear and alternating, then for any σ S n, f(v σ(1),..., v σ(n) ) = sgn(σ)f(v 1,..., v n ). Theorem : Let v 1,..., v n be a basis of V. There exists a unique multilinear alternating function f : V n F with f(v 1,..., v n ) = 1. Fromula for f(u 1,..., u n ) in terms of the coordinates of u i s in basis (v 1,..., v n ). Theorem : Let f : V n F be multilinear and alternating, v 1,..., v n a basis of V, and f(v 1,..., v n ) 0. Then any u 1,..., u n V are linearly independent if and only if f(u 1,..., u n ) 0. Determinant of a matrix as the unique multilinear alternating function of rows det : (F n ) n F with det(e 1,..., e n ) = 1. Formulas for determinants for n = 2, 3. Cor : A is invertible iff det(a) 0. Thm : det(ab) = det(a) det(b). Prop (Further properties of determinants): (a) det(a) is the unique alternating multilinear function of columns of A such that det(i n ) = 1; (b) determinant does not change if a row (column) multiplied by a scalar is added to another row (column); (c) determinant after multiplying a row (column) by a scalar, det(αa) = α n det(a); (d) determinant after swapping two rows or two columns; (e) determinant of a block matrix. Theorem (Laplace expansion): For any i, j {1,..., n}, (a) det(a) = n k=1 ( 1)k+j α kj deta(k j) (expansion along j-th column), (b) det(a) = n k=1 ( 1)i+k α ik deta(i k) (expansion along i-th row). Inverse matrix expressed by determinant and determinants of the minors. Cramer s rule for solving systems of linear equations. 2 Change of basis, eigenvalues and eigenvectors Change of basis. Transition matrix Q from (e 1,..., e n ) to (e 1,..., e n): i th column of Q as coordinates of e i in basis (e 1,..., e n ), or (e 1,..., e n) = (e 1,..., e n ) Q. Transition matrix Q as the matrix of the identity map from F n with basis (e 1,..., e n) to F n with basis (e 1,..., e n ). 2
3 Prop 1.2.1: If Q is the transition matrix from (e 1,..., e n ) to (e 1,..., e n), then Q 1 is the transition matrix from (e 1,..., e n) to (e 1,..., e n ). Thm (change of basis): Let T : U V be a linear map. If (e 1,..., e n), (e 1,..., e n ) are two bases of U with transision matrix Q from e to e and (f 1,..., f m), (f 1,..., f m ) are two bases of V with transision matrix P from f to f, then A (e,f ) T = P 1 A (e,f) T Q. Cor 1.2.3: Determinant Det T = Det A T does not depend on the choice of basis. Similar matrices. Eigenvectors and eigenvalues of linear operators. Spectrum of linear operator. Characteristic polynomial. Prop 1.2.4: Independence of characteristic polynomial of basis. Theorem 1.2.5: λ 0 Spec T iff P T (λ 0 ) = 0, i.e., eigenvalues are roots of the characteristic polynomial. Theorem 1.2.6: Eigenvectors corresponding to different eigenvalues are linearly independent. Simple spectrum. Prop 1.2.7: If T : U U has a simple spectrum, then there exists a basis e 1,..., e n U (formed by eigenvectors) such that A T = diag(λ 1,..., λ n ). One says that T is diagonalizable. Diagonalizable matrices. Examples of non-diagonalizable operators: (a) characteristic polynomial ( ) can be resolved not in every field, algebraically closed fields, (b) A T = Algebraic and geometric multiplicities of eigenvalues. Eigenspace. Prop 1.2.8: (a) The sum of algebraic multiplicities is n; (b) Geometric multiplicity is at most algebraic multiplicity. Theorem 1.2.9: An operator (resp., its matrix) is diagonalizable if and only if for every eigenvalue of T, its geometric and algebraic multiplicities coincide. Remark: Jordan form of a matrix. 3 Inner product spaces Scalar product in R 2 /R 3. Prop 1.3.1: (a) (u, v) = (v, u); (b) (u, u) = u 2 > 0 if u 0; (c) (u, v) = 0 iff u v or one of them is 0; (d) linearity. Orthonormal basis. Thm 1.3.2: Expressing the scalar product, lengths of vectors and angles between vectors in terms of the coordinates of the vectors. Orientation of a basis. Vector product in R 3. Prop 1.3.3: (a) u v = v u; (b) u v = 0 iff u and v are collinear; (c) linearity. Thm 1.3.4: Expressing the vector product in terms of the coordinates of the vectors in a right-hand oriented basis. Definition of the inner product space. Real and unitary spaces. Prop 1.3.5: (a) αu, βv = α β u, v ; (b) u, v 1 + v 2 = u, v 1 + u, v 2 ; (c) n i=1 α iu i, n j=1 β jv j = n i,j=1 α β i j u i, v j ; (d) 0, v = u, 0 = 0. Examples: R n,c n, C([0, 1], C). In a finite-dimensional vector space, one can always define an inner product. 3
4 Norm of a vector. Prop 1.3.6: (a) u 0 with u = 0 u = 0; (b) αu = α u ; (c) Cauchy-Schwarz inequality. Cauchy-Schwarz inequality in R n, C n and C([0, 1], C). Angle between two vectors in a real inner product space. Prop (Triangle inequality for norms): u + v u + v. Normed vector space. Distance between two vectors. Prop 1.3.8: Properties of the distance. Metric space. Orthogonality of two vectors, u v. Prop 1.3.9: Properties of. Orthogonal system of vectors. Orthonormal system of vectors. Theorem : Every orthogonal system is linearly independent. Corollary : If dim(v ) = n, then every orthogonal system contains at most n vectors. Thm (Gram-Schmidt orthogonalization): If u 1,..., u n V are linearly independent, then there exist v 1,..., v n V orthonormal such that for all k n, Span{v 1,..., v k } = Span{u 1,..., u k }. Corollary : Every finite-dimensional inner product space has an orthonormal basis. Prop (Properties of ONBs): If e 1,..., e n is an ONB of V, then (a) u V!u 1,..., u n F such that u = u 1 e u n e n, and u i = u, e i, (b) for all u = u 1 e u n e n, v = v 1 e v n e n V, u, v = u 1 v u n v n, (c) u 2 = u u n 2 (Parseval s identity). Sum of vector subspaces and direct sum of vector subspaces. Prop : (a) U 1 +U 2 is a vector subspace; (b) If V = U 1 U 2, then v V!v 1 U 1, v 2 U 2 such that v = v 1 + v 2. Orthogonal complement of a set = S V, S. Prop (Properties of ): (a) { 0 } = V, V = { 0 }, (b) S is a vector subspace of V (even if S is not), (c) S = (SpanS), (d) (S ) = SpanS; in particular, if S is a vector subspace, then (S ) = S. Theorem : If U is a vector subspace of V (V is finite dimensional), then V = U U. Corollary : Any vector v V can be decomposed uniquely as v = v U + v U, where v U U and v U U. Orthogonal projection, P U. Prop (Properties of P U ): (a) P U is linear, (b) P U U = Id, (c) PU 2 = P U, (d) for all v 1, v 2 V, P U (v 1 ), v 2 = v 1, P U (v 2 ), (e) KerP U = U, ImP U = U. Example (projection on the line spanned by vector u): If U = Span{u}, then P U (v) = v,u u. u 2 Distance between sets S 1, S 2 V, ρ(s 1, S 2 ). Proposition : If U is a vector subspace of V and v V, then ρ(v, U) = v U = P U (v). Dual space or the space of linear functionals on V, V = L(V, F ). Example: for any u V, the map f u : V F defined by f u (v) = v, u is in V. Theorem (Riesz representation theorem): If V is a finite dimensional inner product space, then for any f V there exists unique u V such that f = f u, i.e., for all v V, f(v) = v, u. 4
5 Theorem : For any linear map T : V V there exists unique linear map T : V V such that for all u, v V, T (u), v = u, T (v). T is the adjoint operator of T. Properties : (a) (T ) = T, (b) (αt ) = αt, (c) (T 1 + T 2 ) = T 1 + T 2, (d) (T 1 T 2 ) = T 2 T 1, (e) if e 1,..., e n is an ONB of V, then A T = (A T ) t. Self-adjoint operator, T = T. Example: P U. Hermitian matrix, A t = A, symmetric matrix, A t = A. Proposition : T is self-adjoint iff its matrix is Hermitian in any ONB. Properties: If T 1, T 2 are self-adjoint then T 2 1, T 1 + T 2, αt 1, T 1 T 2 + T 2 T 1 are also self-adjoint, but T 1 T 2 is not necessarily self-adjoint. Theorem : (a) All eigenvalues of a self-adjoint operator are real. In particular, characteristic polynomial is real. (b) Eigenvectors corresponding to different eigenvalues are othogonal. (c) T is self-adjoint iff there exists an ONB in which A T is diagonal and real. In particular, symmetric matrices are diagonalizable. Remark: Normal operator. Unitary operator, T = T 1. Unitary operators in real spaces are called orthogonal. Unitary operators are normal. Proposition : The following conditions are equivalent: (a) T is unitary; (b) for all u, v V, T (u), T (v) = u, v (c) T maps some (or all) ONB to ONB. Unitary matrix, A t = A 1, orthogonal matrix, A t = A. Proposition : T is unitary iff in some ONB A T is unitary. Proposition : If A is unitary, then its rows form an ONB in the space of rows C n, and its columns form an ONB in C n. Proposition : (a) If T is normal and λ is eigenvalue of T with eigenvector u, then λ is eigenvalue of T with the same eigenvector u. (b) If T is unitary, then all its eigenvalues have modulus 1, and deta T = 1. Remark: For any operator T on a vector space over C, det A T equals to the product of all eigenvalues of T (counting algebraic multiplicity). Theorem : (a) If T is unitary, then there exists an ONB such that A T is diagonal with all entries on the diagonal being of modulus 1. (b) If T is orthogonal, then there exists an ONB such that A T is ( block diagonal with) blocks of size 1 being cos ϕi sin ϕ either 1 or 1 and blocks of size 2 being i. (Canonical form of sin ϕ i cos ϕ i unitary/orthogonal operator.) Corollary : Every unitary (resp., orthogonal) matrix is unitary (resp., orthogonally) equivalent to a unitary (resp., orthogonal) matrix in the canonical form. Example: List of all 6 canonical forms of orthogonal operators in R 3. (These are the only transformations of R 3 which preserve lengths and angles between vectors.) 4 Bilinear forms Definition of bilinear form, B(, ). Examples: (a) B(u, v) = n i,j=1 a i,ju i v j, u, v F n, (b) B(f, g) = 1 K(t)f(t)g(t)dt, K, f, g C[0, 1], (c) B(u, v) = f(u)g(v), u, v 0 5
6 V, f, g V, (d) inner product in Euclidean space. Coordinate representation: If e 1,..., e n is a basis of V, then B(u, v) = n i,j=1 u iv j B(e i, e j ). Gram matrix of B: A B = (B(e i, e j )) n i,j=1. Proposition (Change of basis): if (e 1... e n) = (e 1... e n ) C then A B = Ct A B C. Congruent matrices. Rank of a bilinear form, non-degenerate bilinear form. Symmetric bilinear form, B(u, v) = B(v, u). Example. Proposition 1.4.2: If B is symmetric then its matrix in any basis is symmetric, A B = A t B. Theorem (normal form of a symmetric bilinear form): If B is a symmetric bilinear form in a real vector space V, then there exists a basis of V in which the matrix of B is diagonal with only 1 s, 1 s, and 0 s on the diagonal, namely, there exists 1 i = j s 1 s < i = j r e 1,..., e n basis of V and s r n such that B(e i, e j ) =. In 0 i = j > r 0 i j this basis B(u, v) = u 1 v u s v s u s+1 v s+1... u r v r. Proposition (Sylvester s law of intertia): The number of 1 s, 1 s, and 0 s in Theorem is independent of the basis in which the matrix is of the above form. Number of 1 s, 1 s and 0 s form the signature of B. Quadratic form associated to a symmetric bilinear form B, B(v, v). Theorem (polarization of a quadratic form): Any quadratic form in a real vector space is associated to a unique symmetric bilinear form, B(u, v) = (B(u + v, u + v) B(u, u) B(v, v)). Example. 1 2 Theorem 1.4.6: For every quadratic form B(v, v) in R n, there exists a basis in which B(v, v) = v v 2 s v 2 s+1 v 2 r. Numbers s and r are given uniquely by the quadratic form. Lagrange algorithm to find this form ( fill into square ). Positive definite quadratic form, B(v, v) > 0 for v 0. B is positive definite quadratic form on V iff its polarization B defines an inner product on V. Example: Minkowski space, B(v, v) = v v v 2 3 v Functions of several variables Definition of y = f(x 1,..., x n ). Other notation: z = f(x, y), w = f(x, y, z). x = x x 2 n. Open ball B r (x). Limit of f at a R n, lim x a f(x), for f : D R and D R n such that B r (a) D contains points other than a for any r > 0. (Mind: f(a) may be undefined.) Properties of the limit. Examples: lim (x,y) (0,0) estimates). x 2 y x 2 +y 2 = 0, lim (x,y) (0,0) x 3/2 y x 2 +y 2 = 0 (proof from definition/by Proposition 2.1.1: Let α 1 = α 1 (t),..., α n = α n (t) be real functions defined on an interval containing t 0 R, and let α(t) = (α 1 (t),..., α n = α n (t)). If 6
7 for all i {1,..., n}, lim t t0 α i (t) = a i, for some r > 0 and all t (t r, t + r) \ {t 0 }, α(t) a and lim x a f(x) = L, then lim t t0 f(α 1 (t),..., α n (t)) = L. (This is useful in proving non-existence of the limit: if for two different choices of α i s, the corresponding limits lim t t0 f(α 1 (t),..., α n (t)) are different, then the limit of f at a does not exist.) Examples: (a) f(x, y) = xy, for x 2 + y 2 > 0 (limit at 0 does not exist), (b) x 2 +y 2 f(x, y) = x2 y, for x 2 + y 2 > 0 (limit at 0 along every line is 0, but the limit at 0 x 4 +y 2 does not exist). Proposition 2.1.2: If f(x 1,..., x n ) = g( x ) for some g : R + 0 R, then lim x 0 f(x) = lim r 0+ g(r). Such f s are called isotropic. Example: lim (x,y) (0,0) (x 2 + y 2 ) ln(x 2 + y 2 ) = lim r 0+ r 2 ln(r 2 ) = 0. Continuous function at a R n. Continuous function on D. Examples: (a) polynomials, (b) rational functions (in their domains). Open set, limit points, and closed sets. Examples: B r (x) is open but not closed, B r (x) and S r (x) are closed but not open, and R n are both open and closed. Complement of a set. Proposition 2.1.3: D R n is closed if and only if D c is open. Closure of a set. Remark: D is closed; D it is the smallest closed set containig D; D = D. Bounded set. Theorem 2.1.4: If a function is continuous on a bounded and closed set D R n, then it is bounded on D and attains its maximal and minimal values at some points of D. Remark: In metric spaces valid for D compact; all compact sets are bounded and closed, but not all bounded and closed sets must be compact. 6 Literature 1. I. Lankham, B. Nachtergaele, A. Schilling. Linear algebra (as an introduction to abstract mathematics). 2. A. Schüler. Calculus. Lecture notes, Leipzig University K. Hoffman and R. Kunze. Linear algebra. 4. G. Fichtenholz. Differential- und Integralrechnung. 5. S. Mac Lane and G. Birkhoff. Algebra. 6. T.L. Chow. Mathematical methods for physicists: A concise introduction. 7. G. Knieper. Mathematik für Physiker. 7
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
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 informationPh.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system
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 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 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
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 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 information1. Foundations of Numerics from Advanced Mathematics. Linear Algebra
Foundations of Numerics from Advanced Mathematics Linear Algebra Linear Algebra, October 23, 22 Linear Algebra Mathematical Structures a mathematical structure consists of one or several sets and one or
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations
More informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
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 informationLINEAR 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
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 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 informationBASIC 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,,
More informationA Brief Outline of Math 355
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
More information6 Inner Product Spaces
Lectures 16,17,18 6 Inner Product Spaces 6.1 Basic Definition Parallelogram law, the ability to measure angle between two vectors and in particular, the concept of perpendicularity make the euclidean space
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationIntroduction to Linear Algebra, Second Edition, Serge Lange
Introduction to Linear Algebra, Second Edition, Serge Lange Chapter I: Vectors R n defined. Addition and scalar multiplication in R n. Two geometric interpretations for a vector: point and displacement.
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 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 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
More informationLinear Algebra Lecture Notes-II
Linear Algebra Lecture Notes-II Vikas Bist Department of Mathematics Panjab University, Chandigarh-64 email: bistvikas@gmail.com Last revised on March 5, 8 This text is based on the lectures delivered
More informationLast name: First name: Signature: Student number:
MAT 2141 The final exam Instructor: K. Zaynullin Last name: First name: Signature: Student number: Do not detach the pages of this examination. You may use the back of the pages as scrap paper for calculations,
More informationLinear 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
More informationDiagonalizing Matrices
Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n non-singular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B
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 informationVector Spaces, Affine Spaces, and Metric Spaces
Vector Spaces, Affine Spaces, and Metric Spaces 2 This chapter is only meant to give a short overview of the most important concepts in linear algebra, affine spaces, and metric spaces and is not intended
More informationNumerical Linear Algebra
University of Alabama at Birmingham Department of Mathematics Numerical Linear Algebra Lecture Notes for MA 660 (1997 2014) Dr Nikolai Chernov April 2014 Chapter 0 Review of Linear Algebra 0.1 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 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 informationMATH 304 Linear Algebra Lecture 34: Review for Test 2.
MATH 304 Linear Algebra Lecture 34: Review for Test 2. Topics for Test 2 Linear transformations (Leon 4.1 4.3) Matrix transformations Matrix of a linear mapping Similar matrices Orthogonality (Leon 5.1
More informationLinear 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......................
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 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
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 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 informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
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 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 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 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
More informationLinear algebra 2. Yoav Zemel. March 1, 2012
Linear algebra 2 Yoav Zemel March 1, 2012 These notes were written by Yoav Zemel. The lecturer, Shmuel Berger, should not be held responsible for any mistake. Any comments are welcome at zamsh7@gmail.com.
More informationChapter 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
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 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 informationLinear 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
More informationDS-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
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationLinear 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
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationLinGloss. A glossary of linear algebra
LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal
More informationMath 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,...,
More informationELEMENTARY MATRIX ALGEBRA
ELEMENTARY MATRIX ALGEBRA Third Edition FRANZ E. HOHN DOVER PUBLICATIONS, INC. Mineola, New York CONTENTS CHAPTER \ Introduction to Matrix Algebra 1.1 Matrices 1 1.2 Equality of Matrices 2 13 Addition
More informationOptimization Theory. A Concise Introduction. Jiongmin Yong
October 11, 017 16:5 ws-book9x6 Book Title Optimization Theory 017-08-Lecture Notes page 1 1 Optimization Theory A Concise Introduction Jiongmin Yong Optimization Theory 017-08-Lecture Notes page Optimization
More informationLinear Algebra problems
Linear Algebra problems 1. Show that the set F = ({1, 0}, +,.) is a field where + and. are defined as 1+1=0, 0+0=0, 0+1=1+0=1, 0.0=0.1=1.0=0, 1.1=1.. Let X be a non-empty set and F be any field. Let X
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 informationMATH 115A: SAMPLE FINAL SOLUTIONS
MATH A: SAMPLE FINAL SOLUTIONS JOE HUGHES. Let V be the set of all functions f : R R such that f( x) = f(x) for all x R. Show that V is a vector space over R under the usual addition and scalar multiplication
More informationLinear Algebra. Workbook
Linear Algebra Workbook Paul Yiu Department of Mathematics Florida Atlantic University Last Update: November 21 Student: Fall 2011 Checklist Name: A B C D E F F G H I J 1 2 3 4 5 6 7 8 9 10 xxx xxx xxx
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 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: Kolman-Hill 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 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 informationMATH 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
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 informationMath 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
More informationLAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM
LAKELAND COMMUNITY COLLEGE COURSE OUTLINE FORM ORIGINATION DATE: 8/2/99 APPROVAL DATE: 3/22/12 LAST MODIFICATION DATE: 3/28/12 EFFECTIVE TERM/YEAR: FALL/ 12 COURSE ID: COURSE TITLE: MATH2800 Linear Algebra
More informationMATH 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
More informationLINEAR ALGEBRA 1, 2012-I PARTIAL EXAM 3 SOLUTIONS TO PRACTICE PROBLEMS
LINEAR ALGEBRA, -I PARTIAL EXAM SOLUTIONS TO PRACTICE PROBLEMS Problem (a) For each of the two matrices below, (i) determine whether it is diagonalizable, (ii) determine whether it is orthogonally diagonalizable,
More informationHonors Algebra II MATH251 Course Notes by Dr. Eyal Goren McGill University Winter 2007
Honors Algebra II MATH251 Course Notes by Dr Eyal Goren McGill University Winter 2007 Last updated: April 4, 2014 c All rights reserved to the author, Eyal Goren, Department of Mathematics and Statistics,
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 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 information(f + g)(s) = f(s) + g(s) for f, g V, s S (cf)(s) = cf(s) for c F, f V, s S
1 Vector spaces 1.1 Definition (Vector space) Let V be a set with a binary operation +, F a field, and (c, v) cv be a mapping from F V into V. Then V is called a vector space over F (or a linear space
More informationLecture 1: Review of linear algebra
Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations
More informationMath 554 Qualifying Exam. You may use any theorems from the textbook. Any other claims must be proved in details.
Math 554 Qualifying Exam January, 2019 You may use any theorems from the textbook. Any other claims must be proved in details. 1. Let F be a field and m and n be positive integers. Prove the following.
More informationQuizzes for Math 304
Quizzes for Math 304 QUIZ. A system of linear equations has augmented matrix 2 4 4 A = 2 0 2 4 3 5 2 a) Write down this system of equations; b) Find the reduced row-echelon form of A; c) What are the pivot
More informationMath 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
More informationMTH 2032 SemesterII
MTH 202 SemesterII 2010-11 Linear Algebra Worked Examples Dr. Tony Yee Department of Mathematics and Information Technology The Hong Kong Institute of Education December 28, 2011 ii Contents Table of Contents
More informationPRACTICE PROBLEMS FOR THE FINAL
PRACTICE PROBLEMS FOR THE FINAL Here are a slew of practice problems for the final culled from old exams:. Let P be the vector space of polynomials of degree at most. Let B = {, (t ), t + t }. (a) Show
More informationReview of some mathematical tools
MATHEMATICAL FOUNDATIONS OF SIGNAL PROCESSING Fall 2016 Benjamín Béjar Haro, Mihailo Kolundžija, Reza Parhizkar, Adam Scholefield Teaching assistants: Golnoosh Elhami, Hanjie Pan Review of some mathematical
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationand let s calculate the image of some vectors under the transformation T.
Chapter 5 Eigenvalues and Eigenvectors 5. Eigenvalues and Eigenvectors Let T : R n R n be a linear transformation. Then T can be represented by a matrix (the standard matrix), and we can write T ( v) =
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationApplied Linear Algebra in Geoscience Using MATLAB
Applied Linear Algebra in Geoscience Using MATLAB Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots Programming in
More information4. 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.
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 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 informationMATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS
MATH 320: PRACTICE PROBLEMS FOR THE FINAL AND SOLUTIONS There will be eight problems on the final. The following are sample problems. Problem 1. Let F be the vector space of all real valued functions on
More informationMAT 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
More informationLINEAR ALGEBRA REVIEW
LINEAR ALGEBRA REVIEW JC Stuff you should know for the exam. 1. Basics on vector spaces (1) F n is the set of all n-tuples (a 1,... a n ) with a i F. It forms a VS with the operations of + and scalar multiplication
More informationMath 315: Linear Algebra Solutions to Assignment 7
Math 5: Linear Algebra s to Assignment 7 # Find the eigenvalues of the following matrices. (a.) 4 0 0 0 (b.) 0 0 9 5 4. (a.) The characteristic polynomial det(λi A) = (λ )(λ )(λ ), so the eigenvalues are
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
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 informationTranspose & Dot Product
Transpose & Dot Product Def: The transpose of an m n matrix A is the n m matrix A T whose columns are the rows of A. So: The columns of A T are the rows of A. The rows of A T are the columns of A. Example:
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 informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationMath 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
More informationTopic 2 Quiz 2. choice C implies B and B implies C. correct-choice C implies B, but B does not imply C
Topic 1 Quiz 1 text A reduced row-echelon 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 correct-choice may have 0, 1, 2, or 3 choice may have 0,
More information235 Final exam review questions
5 Final exam review questions Paul Hacking December 4, 0 () Let A be an n n matrix and T : R n R n, T (x) = Ax the linear transformation with matrix A. What does it mean to say that a vector v R n is an
More informationLINEAR 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,
More information