Linear Equations in Linear Algebra

Transcription

1 1 Linear Equations in Linear Algera 1.4 THE MATRIX EQUATION A

2 MATRIX EQUATION A mn Definition: If A is an matri, with columns a 1,, a n, and if is in R n, then the product of A and, denoted y A, is the linear comination of the columns of A using the corresponding entries in as weights; that is, 1 A n 2 a a a a a... a 1 2 n n n Note that A is defined only if the numer of columns of A equals the numer of entries in.. Slide 1.4-2

3 MATRIX EQUATION A Eample 2: For v 1, v 2, v 3 in R m, write the linear comination 3v 5v 7v as a matri times a vector. Solution: Place v 1, v 2, v 3 into the columns of a matri A and place the weights 3, 5, and 7 into a vector. That is, 3 3v 5v 7v v v v 5 A Slide 1.4-3

4 MATRIX EQUATION A Now, write the system of linear equations as a vector equation involving a linear comination of vectors. For eample, the following system is equivalent to (1). (2) Slide 1.4-4

5 MATRIX EQUATION A As in the eample, the linear comination on the left side is a matri times a vector, so that (2) ecomes A. (3) Equation (3) has the form. Such an equation is called a matri equation, to distinguish it from a vector equation such as shown in (2). Slide 1.4-5

6 MATRIX EQUATION A THEOREM 3 If A is an m n matri, with columns a 1,, a n, and if is in R n, then the matri equation A = has the same solution set as the vector equation 1 a a n a n = which, in turn, has the same solution set as the system of linear equations whose augmented matri is [a 1 a 2 a n ] Slide 1.4-6

7 EXISTENCE OF SOLUTIONS A The equation has a solution if and only if is a linear comination of the columns of A. THEOREM 4 Let A e an m n matri. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a. For each in R m, the equation A= has a solution.. Each in R m is a linear comination of the columns of A. c. The columns of A span R m. d. A has a pivot position in every row. Slide 1.4-7

8 COMPUTATION OF A Eample 4: Compute A, where 1 and. 2 3 Solution: From the definition, A Slide 1.4-8

9 COMPUTATION OF A (1) The first entry in the product A is a sum of products (sometimes called a dot product), using the first row of A and the entries in. Slide 1.4-9

10 COMPUTATION OF A That is,. 2 3 Similarly, the second entry in A can e calculated y multiplying the entries in the second row of A y the corresponding entries in and then summing the resulting products Slide

11 ROW-VECTOR RULE FOR COMPUTING A Likewise, the third entry in A can e calculated from the third row of A and the entries in. If the product A is defined, then the ith entry in A is the sum of the products of corresponding entries from row i of A and from the vector. The matri with 1 s on the diagonal and 0 s elsewhere is called an identity matri and is denoted y I. For eample, is an identity matri. Slide

12 PROPERTIES OF THE MATRIX-VECTOR PRODUCT A Theorem 5: If A is an matri, u and v are vectors in R n, and c is a scalar, then a... mn A(u v) Au Av; A( cu) c( Au) n 3 A a a a Proof: For simplicity, take,, and u, v in R 3. i 1,2,3, For let u i and v i e the ith entries in u and v, respectively. Slide

13 PROPERTIES OF THE MATRIX-VECTOR PRODUCT A THEOREM 5 If A is an m n matri, u and v are vectors in R n, and c is a scalar, then a. A(u + v) = Au + Av;. A(cu) = c(au). Proof: For simplicity, take n = 3, A = [a 1 a 2 a 3 ], and u, v in R 3. For i = 1, 2, 3, let u i and v i e the ith entries in u and v, respectively. Slide

14 PROPERTIES OF THE MATRIX-VECTOR PRODUCT A A(u v) To prove statement (a), compute as a linear comination of the columns of A using the entries in u vas weights. u v A(u v) a a a u v u v ( u v )a ( u v )a ( u v )a Entries in u v Columns of A ( u a u a u a ) ( v a v a v a ) AuAv Slide

15 PROPERTIES OF THE MATRIX-VECTOR PRODUCT A Ac ( u) To prove statement (), compute as a linear comination of the columns of A using the entries in cu as weights. cu1 A( cu) a a a cu ( cu )a ( cu )a ( cu )a cu3 c( u a ) c( u a ) c( u a ) c( u a u a u a ) ca ( u) Slide