.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45
Matrix Addition Theorem (pg 0) Let A, B, and C be matrices of the same size, and let r & s be scalars..) A + B B+ A.) (A + B) + C A+ (B+ C).) A + 0 A 4.) r(a + B) ra+ rb 5.) (r + s)a ra+ sa 6.) r(sa) (rs)a Example 6 4 Let A and 0 8 B 5. Compute each of the following: 4 - A B - A A + B 46
MATRIX MULTIPLICATION How do I multiply matrices? Row-Column Rule For Computing AB Theorem (pg 4) Let A be an mxn matrix, and let B & C have sizes for which the indicated sums and products are defined. Let r be any scalar..) A(BC) (AB)C.) A(B + C) AB+ AC.) (B + C)A BA+ CA 4.) r(ab) (ra)b A(rB) 5.) I ma A AIn Associative law of multiplication Left distributive law Right distributive law Identity for matrix multiplication 47
Example 0 5 Let A, B, and C. 4 5 4 Compute each of the following: AC CA BC CB 48
.) In general, AB BA. WARNINGS!.) The cancellation laws do NOT hold for matrix multiplication. That is, if AB AC then it is NOT necessarily true thatb C..) If a product AB is the zero matrix, you CANNOT conclude in general that A 0 or B 0. What is the transpose of a matrix? 49
Theorem (pg 6) Let A and B denote matrices whose sizes are appropriate for the following sums and products and let r be any scalar. T T.) ( A ) A T A + B A + B T T.) ( ) T T.) ( ra ) ra T T 4.) ( AB ) B A T Example 5 Let A and x. Compute each of the following: 4 6 T ( Ax ) x T A T T xx x T x Is A T x T defined? Why or why not. 50
When is a matrix A invertible?.: THE INVERSE OF A MATRIX Singular Matrix vs. Nonsingular Matrix Theorem 4 (CAUTION: THIS ONLY WORKS FOR x MATRICES!) Example Find the inverse (if it exists) of each of the following matrices. 8 4 6 6 4 8 5
Theorem 5 Theorem 6 Theorem 7 Algorithm for Finding A (pg 5).) Set up the augmented matrix [A I]..) Row reduce the matrix into reduced echelon form..) If A is row equivalent to I, then [A I] is row equivalent to [I have an inverse. A ]. Otherwise, A does not 5
5 Example Let 8 0 5 A and 5 4 b. Find A and use it to solve b x A.
Example Let A, B, C, D, X, and Y be invertible nxn matrices. Solve the equation, A ( B CX) D Y + for X. Things to keep in mind: Matrix division does not exist. You cannot divide by a matrix. Also, keep the order of multiplication consistent. If you multiply by A, on the left of the left side of the equation you must multiply by A on the left of the right side of the equation as well. 54
.: CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: The Invertible Matrix Theorem (pg 6) Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false..) A is an invertible matrix..) A is row equivalent to the nxn identity matrix, I n..) A has n pivot positions. 4.) The equation A x 0 has only the trivial solution. 5.) The columns of A form a linearly independent set. 6.) The linear transformation n n T : R R given by T ( x) Ax 7.) The equation A x b has exactly one solution for each b in is one-to-one. n R. 8.) The columns of A span 9.) The linear transformation n R. n n T : R R given by T ( x) Ax 0.) There is an nxn matrix C such that CA AC In. is onto..) T A is an invertible matrix. 55
What is an invertible transformation? Theorem 9 What if T : R n n R is one-to-one? Onto? 56
Example x x x Let T : R R by T x x+ x x be a linear transformation. Show that T is x 6x+ 4x invertible and find T..4 SUBSPACES OF n R 57
What is a vector space? What is a subspace? What is the subspace test? 58
Example Determine which of the following are subspaces of R. 59
Example Use the subspace test to determine if the following is a subspace of x+ y H y x x,y R x+ 4y R. 60
6 Example Use the subspace test to determine if the following is a subspace of R. + R a,b,c c b a c b W
What is ColA? What is NulA? Theorem 0 6
Example 4 Let A. 7 8 5 7 0 What is ColA? ColA is a subspace of What is NulA? k R, what is k in this example? NulA is a subspace of s R, what is s in this example? 6
64 What is a basis? What is the standard basis for n R? Theorem Example 5 Let 0 7 5 8 7 A. Find a basis for ColA and NulA.
65 Example 6 Determine which sets are bases for R or R. Justify each answer.,,,,, 0 0,, 5 7 8
66.5 DIMENSION & RANK What is the dimension of a subspace? Example Determine the dimension of the subspace H of R spanned by the vectors v, v, 7 0 v, 4 6 7 v 4.
What is the rank of a matrix? Theorem (The Rank Theorem) Example Suppose a x5 matrix A has pivot columns. Is ColA R? Is NulA R? Suppose a 4x7 matrix A has pivot columns. Is ColA R? What is the dimension of NulA? 67
Example Construct a 4x matrix with rank. Theorem (The Basis Theorem) 68
69 Example 4 Let 4 5 4 0 A. Is the set 0, 0, 9 0 9 6 S a basis for ColA?
Theorem 8: The Invertible Matrix Theorem (pg 90) Let A be a square nxn matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false..) A is an invertible matrix..) A is row equivalent to the nxn identity matrix, I n..) A has n pivot positions. 4.) The equation A x 0 has only the trivial solution. 5.) The columns of A form a linearly independent set. 6.) The linear transformation n n T : R R given by T ( x) Ax 7.) The equation A x b has exactly one solution for each b in is one-to-one. n R. 8.) The columns of A span 9.) The linear transformation n R. n n T : R R given by T ( x) Ax 0.) There is an nxn matrix C such that CA AC In. is onto..) T A is an invertible matrix..) The columns of A form a basis for.) ColA n R 4.) dim ( ColA) n 5.) rank ( A) n 6.) NulA {0} 7.) dim ( NulA) 0 8.) det A 0 n R. 70