MATH 2331 Linear Algebra. Section 2.1 Matrix Operations. Definition: A : m n, B : n p. Example: Compute AB, if possible.

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1 MATH 2331 Linear Algebra Section 2.1 Matrix Operations Definition: A : m n, B : n p ( 1 2 p ) ( 1 2 p ) AB = A b b b = Ab Ab Ab Example: Compute AB, if possible. 1

2 Row-column rule: i-j-th entry of AB: row i column j 2

3 Properties of Matrix Multiplication: A : m n matrix. If the following are defined- A( BC) = ( AB) C A( B + C) = AB + AC ( B + C) A = BA + CA r( AB) = ( ra) B = A( rb) I A = A = AI m n Question: Is AB = BA? 3

4 Warnings: In general, AB BA If AB = AC, then you can t cancel. If AB = 0, it is possible that A and B are both nonzero. 4

5 Definition: A : m n matrix. The transpose of A is the n m matrix whose columns are formed from the corresponding rows of A. Notation: T A 5

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7 Theorem: Let A and B be matrices so that the following operations are defined. T ( A ) T = A ( A + B) T T = A T + B ( ra) T T = ra ( AB) T T T = B A 6

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10 MATH 2331 Linear Algebra Section 2.2 The Inverse of a Matrix Let A be an n n matrix. Definition: A is said to be invertible if there exists an n n matrix B such that AB = BA = I n. If A is not invertible, then it is said to be singular. 1 If A has an inverse, then the inverse is unique - notation: A. 1

11 2 Let a b A c d =. Theorem: Let a b A c d =. If 0 ad bc, then A is invertible and 1 1 d b A ad bc c a = If 0 ad bc = = = =, then A is not invertible. Example:

12 3 Let A be n n ; Example: If exists, find the inverse of A =

13 Theorem 5: If A is an invertible matrix, then for any b R n, the equation Ax = b has the unique solution 1 =. x A b Proof: FACTS: 1) If A is invertible, then so is 1 A 1. Moreover, ( A ) 1 = A. 4

14 2) If A, B are invertible n n matrices, then AB is invertible. Moreover, ( ) AB = B A 3) If A is invertible, then so is 1 T T 1 A. Moreover, ( A ) ( A ) T =. 5

15 Elementary Matrices An elementary matrix is one that is obtained by performing a single elementary operation on an identity matrix. For example: 6

16 7 Example: a b c A d e f g h i =

17 Fact: Each elementary matrix is invertible. The inverse of E is the elementary matrix of the same type that transforms E back into I. 8

18 Theorem: An n n matrix is invertible if and only if A is row equivalent to I n. Any sequence of row operations that reduces A to I n also transforms I n into 1 A. 9

19 Algorithm: Row-reduce the augmented matrix [ A I ]. If A is row equivalent to I, then [ A I ] is row equivalent to I 1 A. Otherwise, A is not invertible. 10

20 Section 2.3 Characterizations of Invertible Matrices Theorem 8: The Invertible Matrix Theorem Let A be an n n matrix. The following are equivalent: a. A is an invertible matrix. b. A is row-equivalent to the n n identity matrix. c. A has n pivot positions d. The equation 0 Ax = has only the trivial solution. e. The columns of A are linearly independent. f. The linear transformation x g. The equation h. The columns of A span R n. Ax is one-to-one. Ax = b has at least one solution for each b R n. i. The linear transformation x Ax maps R n onto R n. j. There is an nxn matrix C such that CA=I. k. There is an nxn matrix D such that AD=I l. T A is invertible. 11

21 12 Two tests to check whether a matrix is invertible or not: Test 1: The only solution to the system Ax=[0] is [0]. Test 2: Reduced form has n pivots ( full pivots). Example: Is the following matrix invertible? A =

22 13 Example: Invertible? A = B = C = D = E =

23 F = G = Example: Let A be an n n upper triangular matrix. Is it invertible? FACT:

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29 Definition: A linear transformation :R n n T R is invertible if there exists a linear transformation : R n n S R such that S( T( x)) = T( S( x)) = x x. = =, for all R n 1 S is the inverse of T, notation: T. Theorem: Let induced by T. :R n n T R be a linear transformation and A be the standard matrix T is invertible if and only if A is an invertible matrix. 1 1 In this case, ( ) T x = A x. 15

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31 2 2 Example: T :R R ; x1 x1 + 2x2 T = x 2 4x1 + 6x2 Is T invertible? If yes, find the inverse. 16

32 Example: Let T be a one-to-one and onto linear transformation. Is it invertible? Example: Let T be an invertible linear transformation. Is it one-to-one? Is it onto? 17

33 Exercise: Let A,B be n n matrices. Show that if AB is invertible, then A is invertible. 18

34 Extra Ex: Let A be an nxn matrix. True or False? If the equation Ax = 0 has only the trivial solution, then A is row equivalent to I. If the columns of A span R n, then the columns are linearly independent. Ax=b is consistent for any b in R n. If Ax=0 has nontrivial solutions, then A has less than n pivots. If the columns of A are linearly independent, then the columns span R n. If the equation Ax=b is consistent for each b in R n, then the transformation is one-to-one. x Ax If Ax=b has infinitely many solutions for some b in R n, then A is not invertible. 19

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36 MATH 2331 Linear Algebra Section 2.4 Partitioned matrices 1

37 Fact: T T T A B A C = T T C D B D Fact: A block diagonal matrix is invertible if each block on the diagonal is invertible. A 0 0 B is invertible if A and B are invertible. 2

38 Section 2.8, 2.9: Basis, Dimension, Rank, Column Space, Nullspace. Definition: The column space of a matrix A is the set of all linear combinations of the columns of A. Notation: Col(A) The null space of a matrix A is the set of all solutions to Ax = 0. Notation: Null(A) Rank of a matrix A is the number of pivot columns when the matrix is reduced. Notation: rank(a) 3

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40 Definition: A basis for a subspace H is a linearly independent set that spans H. The dimension of a subspace is the number of elements in the basis. Fact: The pivot columns of A form a basis for Col(A). Fact: To find a basis for Null(A), describe the solution of Ax=0 in parametric form. 4

41 Invertible Matrix Theorem (continued) A: nxn matrix. A is invertible if and only if n m. The columns of A form a basis for R n. Col( A ) =R o. dim( Col( A)) p. rank( A) n = n q. Null( A ) = {0} = n r. dim( Null( A )) = 0 5

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2. Every linear system with the same number of equations as unknowns has a unique solution.

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