Linear Algebra

198 Chapter Three. Maps Between SpacesWe can also view the operation column-by-column.⎛⎞ ⎛ ⎞ ⎛⎞h 1,1 h 1,2 . . . h 1,n c 1 h 1,1 c 1 + h 1,2 c 2 + · · · + h 1,n c nh 2,1 h 2,2 . . . h 2,nc 2 ⎜ .⎟ ⎜⎝ .⎠ ⎝. ⎟⎠ = h 2,1 c 1 + h 2,2 c 2 + · · · + h 2,n c n⎜⎟⎝.⎠h m,1 h m,2 . . . h m,n c n h m,1 c 1 + h m,2 c 2 + · · · + h m,n c n⎛ ⎞ ⎛ ⎞h 1,1h 1,nh 2,1= c 1 ⎜ ⎟⎝ . ⎠ + · · · + c h 2,nn ⎜ ⎟⎝ . ⎠h m,1 h m,n1.11 Example( ) ⎛ ⎞2 ( ) ( ) ( ) ( )1 0 −1 ⎜ ⎟ 1 0 −1 1⎝ −1⎠ = 2 − 1 + 1 =2 0 32 0 3 71The result has the columns of the matrix weighted by the entries of the vector.This way of looking at it brings us back to the objective stated at the start ofthis section, to compute h(c 1⃗β 1 + · · · + c n⃗β n ) as c 1 h(⃗β 1 ) + · · · + c n h(⃗β n ).We began this section by noting that the equality of these two enables us tocompute the action of h on any argument knowing only h(⃗β 1 ), . . . , h(⃗β n ). Wehave developed this into a scheme to compute the action of the map by takingthe matrix-vector product of the matrix representing the map with the vectorrepresenting the argument. In this way, with respect to any bases, any linearmap has a matrix representing it. The next subsection will show the converse,that if we fix bases then for any matrix there is an associated linear map.Exerciseš 1.12 Multiply the matrixby each vector (or state “not defined”).⎛ ⎞⎛ ⎞2 ( ) 0(a) ⎝1⎠−2(b) (c) ⎝0⎠−200⎛1 3⎞1⎝0 −1 2⎠1 1 01.13 Perform, if possible, each matrix-vector multiplication.( ( ) ( ) 2 1 4 1 1 0(a)(b)3 −1/2) ⎛ ⎞⎝1⎠ (c)2 −2 1 0̌ 1.14 Solve this matrix equation.⎛2 1⎞ ⎛ ⎞1 x⎛ ⎞8⎝0 1 3⎠⎝y⎠ = ⎝4⎠1 −1 2 z 431( 1 1−2 1) ⎛ ⎞⎝13⎠1