Ax equal to b

Ax equal to b - 1.4 The Matrix Equation Ax  b Linear...

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Unformatted text preview: 1.4 The Matrix Equation Ax  b Linear combinations can be viewed as a matrix-vector multiplication. Definition If A is an m n matrix, with columns a 1 , a 2 , , a n , and if x is in R n , then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding entries in x as weights. I.e., x1 Ax  x2 a1 a2  an   x1 a1  x2 a2    xnan xn EXAMPLE: 1 4 3 2 0 5 7 6 7 21 0 1 7 4 6 3 0 12 30  5 24  2 31  9 30 1 EXAMPLE: Write down the system of equations corresponding to the augmented matrix below and then express the system of equations in vector form and finally in the form Ax  b where b is a 3 1 vector. 234 9 310 2 Solution: Corresponding system of equations (fill-in) Vector Equation: 2 3  3 1  4 0  9 2 . Matrix equation (fill-in): 2 Three equivalent ways of viewing a linear system: 1. as a system of linear equations; 2. as a vector equation x 1 a 1  x 2 a 2    x n a n  b; or 3. as a matrix equation Ax  b. THEOREM 3 If A is a m n matrix, with columns a 1 , R m , then the matrix equation Ax  b , a n , and if b is in has the same solution set as the vector equation x1a1  x2a2    xnan  b which, in turn, has the same solution set as the system of linear equations whose augmented matrix is a1 a2  an b . Useful Fact: The equation Ax  b has a solution if and only if b is a __________ ________________ of the columns of A. 3 1 5 3 11 14 2 EXAMPLE: Let A  4 8 b1 10 and b  b2 . b3 Is the equation Ax  b consistent for all b? Solution: Augmented matrix corresponding to Ax  b: 1 4 5 b1 3 11 14 b 2 2 8 10 b3 145  b1 011 3b 1  b 2 000 2b 1  b 3 Ax  b is _______ consistent for all b since some choices of b make 2b 1  b 3 nonzero. 4 1 5 3 11 14 2 A 4 8 10 a1 a2 a3 The equation Ax  b is consistent if 2b 1  b 3  0. (equation of a plane in R 3 ) x 1 a 1  x 2 a 3  x 3 a 3  b if and only if b 3 2b 1  0. b3 b2 b1 Columns of A span a plane in R 3 through 0 5 Instead, if any b in R 3 (not just those lying on a particular line or in a plane) can be expressed as a linear combination of the columns of A, then we say that the columns of A span R 3 . Definition We say that the columns of A  span a1 a2  ap R m if every vector b in R m is a linear combination of a1, , ap (i.e. Span a 1 , , a p  R m ). THEOREM 4 Let A be an m n matrix. Then the following statements are logically equivalent: a. For each b in R m , the equation Ax  b has a solution. b. Each b in R m is a linear combination of the columns of A. c. The columns of A span R m . d. A has a pivot position in every row. 6 Proof (outline): Statements (a), (b) and (c) are logically equivalent. To complete the proof, we need to show that (a) is true when (d) is true and (a) is false when (d) is false. Suppose (d) is _____________. Then row-reduce the augmented matrix A b : A b ßß U d and each row of U has a pivot position and so there is no pivot in the last column of U d . So (a) is _____________. Now suppose (d) is _____________. Then the last row of U d contains all zeros. Suppose d is a vector with a 1 as the last entry. Then U d represents an inconsistent system. Row operations are reversible: U d ßß A b Í A b is inconsistent also. So (a) is ____________.  7 12 EXAMPLE: Let A  34 b1 and b  b2 . Is the 56 b3 equation Ax  b consistent for all possible b? Solution: A has only _____ columns and therefore has at most _____ pivots. Since A does not have a pivot in every _______, Ax  b is ________ ___________________ for all possible b, according to Theorem 4. 8 123 EXAMPLE: Do the columns of A  246 span R 3 ? 039 Solution: 123 246  039 (no pivot in row 2) By Theorem 4, the columns of A ______________________________________________. Another method for computing Ax Read Example 4 on page 44 through Example 5 on page 45 to learn this rule for computing the product Ax. Theorem 5 If A is an m n matrix, u and v are vectors in R n , and c is a scalar, then: a. A u  v  Au  Av; b. A cu  cAu. 9 ...
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This note was uploaded on 03/01/2012 for the course EE 101 taught by Professor Munk during the Spring '12 term at Alaska Anch.

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