Linear Algebra

214 Chapter Three. Maps Between Spaces2.3 Definition The matrix-multiplicative product of the m×r matrix G and ther×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 of thefirst matrix with the j-th column of the second.⎛⎞⎛⎞ h 1,j⎛⎞.. . . h 2,j . . .GH = ⎜⎝g i,1 g i,2 . . . g i,r⎟⎠ ⎜⎟⎝ . ⎠ = .⎜⎝. . . p i,j . . . ⎟⎠..h r,j2.4 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 5⎞⎟⎝0 · 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⎞⎛( )⎟ 1 3 ⎜2 · 1 + 0 · 5 2 · 3 + 0 · 7⎞ ⎛ ⎞2 6⎟ ⎜ ⎟⎝4 6⎠= ⎝4 · 1 + 6 · 5 4 · 3 + 6 · 7⎠ = ⎝34 54⎠5 78 28 · 1 + 2 · 5 8 · 3 + 2 · 7 18 38We next check that our definition of the matrix-matrix multiplication operationdoes what we intend.2.6 Theorem A composition of linear maps is represented by the matrix productof the representatives.Proof This argument generalizes 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, ofsizes 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 )Distribute and regroup on the v’s.= (g i,1 h 1,1 + g i,2 h 2,1 + · · · + g i,r h r,1 ) · v 1+ · · · + (g i,1 h 1,n + g i,2 h 2,n + · · · + g i,r h r,n ) · v n