Linear Algebra

210 Chapter Three. Maps Between Spaceswhich we recognizing as the result of this matrix-vector product.⎛ ⎞⎛= ⎝ 1 · 4 + 1 · 5 1 · 6 + 1 · 7 1 · 8 + 1 · 9 1 · 2 + 1 · 3⎞ v 10 · 4 + 1 · 5 0 · 6 + 1 · 7 0 · 8 + 1 · 9 0 · 2 + 1 · 3⎠⎜v 2⎟⎝v 1 · 4 + 0 · 5 1 · 6 + 0 · 7 1 · 8 + 0 · 9 1 · 2 + 0 · 33⎠B,D v 4Thus, the matrix representing g◦h has the rows of G combined with the columnsof H.2.3 Definition The matrix-multiplicative product of the m×r matrix G andthe r×n matrix H is the m×n matrix P , wherep i,j = g i,1 h 1,j + g i,2 h 2,j + · · · + g i,r h r,jthat is, the i, j-th entry of the product is the dot product of the i-th row andthe j-th column.⎛⎞ ⎛⎞h⎛⎞1,j.GH = ⎜⎝g i,1 g i,2 . . . g i,r⎟. . . h 2,j . . .⎠ ⎜⎝⎟. ⎠ = .⎜⎝. . . p i,j . . . ⎟⎠.h r,j.2.4 Example The matrices from Example 2.2 combine in this way.⎛⎝ 1 · 4 + 1 · 5 1 · 6 + 1 · 7 1 · 8 + 1 · 9 1 · 2 + 1 · 3⎞ ⎛⎞9 13 17 50 · 4 + 1 · 5 0 · 6 + 1 · 7 0 · 8 + 1 · 9 0 · 2 + 1 · 3⎠ = ⎝5 7 9 3⎠1 · 4 + 0 · 5 1 · 6 + 0 · 7 1 · 8 + 0 · 9 1 · 2 + 0 · 3 4 6 8 22.5 Example⎛⎝ 2 0⎞ ⎛( )4 6⎠1 3= ⎝ 2 · 1 + 0 · 5 2 · 3 + 0 · 7⎞ ⎛4 · 1 + 6 · 5 4 · 3 + 6 · 7⎠ = ⎝ 2 6⎞34 54⎠5 78 28 · 1 + 2 · 5 8 · 3 + 2 · 7 18 382.6 Theorem A composition of linear maps is represented by the matrixproduct of the representatives.Proof. (This argument parallels Example 2.2.) Let h: V → W and g : W → Xbe represented by H and G with respect to bases B ⊂ V , C ⊂ W , and D ⊂ X,of sizes n, r, and m. For any ⃗v ∈ V , the k-th component of Rep C ( h(⃗v) ) ish k,1 v 1 + · · · + h k,n v nand so the i-th component of Rep D ( g ◦ h (⃗v) ) is this.g i,1 · (h 1,1 v 1 + · · · + h 1,n v n ) + g i,2 · (h 2,1 v 1 + · · · + h 2,n v n )+ · · · + g i,r · (h r,1 v 1 + · · · + h r,n v n )D