The Matrix Equation Ax = b

1.4 The Matrix Equation Ax = b Math 2331 – Linear Algebra 1.4 The Matrix Equation Ax = b Math 2331, Linear Algebra 1 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm 1.4 The Matrix Equation Ax = b Matrix-Vector Multiplication Linear Combination of the Columns Matrix Equation Three Equivalent Ways of Viewing a Linear System Existence of Solution Matrix Equation Equivalent Theorem Another method for computing Ax Row-Vector Rule Math 2331, Linear Algebra 2 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix-Vector Multiplication Key Concepts to Master Linear combinations can be viewed as a matrix-vector multiplication. Matrix-Vector Multiplication If A is an m × n matrix, with columns a1 , a2 , . . . , an , and if x is in Rn , 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     x2  Ax = a1 a2 · · · an  .  = x1 a1 + x2 a2 + · · · + xn an  ..  xn Math 2331, Linear Algebra 3 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix-Vector Multiplication: Examples Example        1 −4  1 −4 7  3 2  = 7  3  + −6  2  = −6 0 5 0 5       7 24 31  21  +  −12  =  9  0 −30 −30 Math 2331, Linear Algebra 4 / 15 Definition Theorem Span Rm 1.4 The Matrix Equation Ax = b Matrix-Vector Multiplication: Examples 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.   2 3 4 9 −3 1 0 −2 Solution: Corresponding system of equations (fill-in) Vector Equation:   2 + −3  3 1   + 4 0   = 9 −2  . Matrix equation (fill-in): Math 2331, Linear Algebra 5 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation Three Equivalent Ways of Viewing a Linear System 1 as a system of linear equations; 2 as a vector equation x1 a1 + x2 a2 + · · · + xn an = b; or 3 as a matrix equation Ax = b. Useful Fact The equation Ax = b has a solution if and only if b is a of the columns of A. Math 2331, Linear Algebra 6 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Theorem Theorem If A is a m × n matrix, with columns a1 , . . . , an , and if b is in Rm , then the matrix equation Ax = b has the same solution set as the vector equation x1 a1 + x2 a2 + · · · + xn an = b which, in turn, has the same solution set as the system of linear equations whose augmented matrix is   a1 a2 · · · an b . Math 2331, Linear Algebra 7 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Example Example    4 5 b1 1 Let A =  −3 −11 −14  and b =  b2  . 2 8 10 b3  Is the equation Ax = b consistent for all b? Solution: Augmented matrix corresponding to Ax = b:    1 4 5 b1 1 4 5 b1  −3 −11 −14 b2  ∼  0 1 1 3b1 + b2  0 0 0 −2b1 + b3 2 8 10 b3  Ax = b is consistent for all b since some choices of b make −2b1 + b3 nonzero. Math 2331, Linear Algebra 8 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Example (cont.)   1 4 5 A =  −3 −11 −14  2 8 10 ↑ ↑ ↑ a1 a2 a3 The equation Ax = b is consistent if −2b1 + b3 = 0. (equation of a plane in R3 ) x1 a1 + x2 a3 + x3 a3 = b if and only if b3 − 2b1 = 0. Columns of A span a plane in R3 through 0 Instead, if any b in R3 (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 R3 . Math 2331, Linear Algebra 9 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm m Matrix Equation: Span R Definition   span We say that the columns of A = a1 a2 · · · ap Rm if every vector b in Rm is a linear combination of a1 , . . . , ap (i.e. Span{a1 , . . . , ap } = Rm ). Theorem (4) Let A be an m × n matrix. Then the following statements are logically equivalent: 1 For each b in Rm , the equation Ax = b has a solution. 2 Each b in Rm is a linear combination of the columns of A. 3 The columns of A span Rm . 4 A has a pivot position in every row. Math 2331, Linear Algebra 10 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Proof of Theorem 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] ∼ · · · ∼ [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 Math 2331, Linear Algebra 11 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Proof of Theorem (cont.) 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] ∼ · · · ∼ [A b] =⇒ [A b] is inconsistent also. So (a) is Math 2331, Linear Algebra .  12 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Example Example  1 Let A =  3 5 consistent for Solution:    2 b1 4  and b =  b2 . Is the equation Ax = b 6 b3 all possible b? A has only columns and therefore has pivots. at most Since A does not have a pivot in every is according to Theorem 4. , Ax = b for all possible b, Math 2331, Linear Algebra 13 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Matrix Equation: Example Example   1 2 3 Do the columns of A =  2 4 6  span R3 ? 0 3 9 Solution:   1 2 3  2 4 6 ∼ 0 3 9 (no pivot in row 2) By Theorem 4, the columns of A . Math 2331, Linear Algebra 14 / 15 1.4 The Matrix Equation Ax = b Definition Theorem Span Rm Another method for computing Ax: Row-Vector Rule Another method for computing Ax: Row-Vector Rule Read Example 4 on page 37 through Example 5 on page 38 to learn this rule for computing the product Ax. Theorem If A is an m × n matrix, u and v are vectors in Rn , and c is a scalar, then: 1 A (u + v) = Au + Av; 2 A (cu) = cAu. Math 2331, Linear Algebra 15 / 15