Chapter 9 - Linear Maps from (R^n) to (R^m)

Section 9.3 - Composites of Two Linear Maps and Inverses

Example 9.3.1

T[{x_,y_,z_,w_}]:={x−y+z+w,3x+5 y−2w,x+7z+3w,x+y+4z,6y−9w,x+2z,−8x+y+11 z};

S[{a_,b_,c_,d_,e_,f_,g_}]:={−6a+3c−2d+f−9g,a−5b+c−d+e+f−g,7a+3b−4c+5d−e−f+8g};

Expand[S[T[{x,y,z,w}]]]

(CMat=Transpose[{S[T[{1,0,0,0}]],S[T[{0,1,0,0}]],S[T[{0,0,1,0}]],S[T[{0,0,0,1}]]}])//MatrixForm

(A=Transpose[{T[{1,0,0,0}],T[{0,1,0,0}],T[{0,0,1,0}],T[{0,0,0,1}]}])//MatrixForm

(B=Transpose[{S[{1,0,0,0,0,0,0}],S[{0,1,0,0,0,0,0}],S[{0,0,1,0,0,0,0}],S[{0,0,0,1,0,0,0}],S[{0,0,0,0,1,0,0}],S[{0,0,0,0,0,1,0}],S[{0,0,0,0,0,0,1}]}])//MatrixForm

B.A//MatrixForm

Example 9.3.2

(Subscript[v, 1])={-1,3,1,5};(Subscript[v, 2])={6,-2,0,-4};

(Subscript[v, 3])={9,11,-2,8};(Subscript[v, 4])={-4,-9,3,1};

(Vrow={(Subscript[v, 1]),(Subscript[v, 2]),(Subscript[v, 3]),(Subscript[v, 4])})//MatrixForm

RowReduce[Vrow]//MatrixForm

(Subscript[w, 1])={2,-4,1,0};(Subscript[w, 2])={1,3,-7,2};

(Subscript[w, 3])={-5,6,-2,1};(Subscript[w, 4])={0,7,-2,-10};

(Wrow={(Subscript[w, 1]),(Subscript[w, 2]),(Subscript[w, 3]),(Subscript[w, 4])})//MatrixForm

RowReduce[Wrow]//MatrixForm

(A=Transpose[Wrow].Inverse[Transpose[Vrow]])//MatrixForm

Det[A]

{A.(Subscript[v, 1]),A.(Subscript[v, 2]),A.(Subscript[v, 3]),A.(Subscript[v, 4])}//MatrixForm

Wrow//MatrixForm