How do we know that S must be induced by some matrix B? Functions T:Rn→Rm are called transformations from Rn to Rm A transformation T:Rn→Rn has an inverse if there is some transformation S such that T∘S=S∘T=1Rn Let T=TA:Rn→Rn denote the matrix transformation induced by the n×n matrix A, that is T(x)=Ax We have: 𝕟BAx=S[T(x)]=(S∘T)x=1Rn(x)=x=Inx

Question

How do we know that S must be induced by some matrix B?  Functions T:Rn→Rm are called transformations from Rn to Rm  A transformation T:Rn→Rn has an inverse if there is some transformation S such that T∘S=S∘T=1Rn  Let T=TA:Rn→Rn denote the matrix transformation induced by the n×n matrix A, that is T(x)=Ax  We have:  𝕟BAx=S[T(x)]=(S∘T)x=1Rn(x)=x=Inx

Jude Carpenter · Accepted Answer

Step 1  We know that S is induced by a matrix because it is linear. After all, since T has an inverse it must be bijective, so for all a,b and some c,d we have  S(a+b)=S(T(c)+T(d))=S(T(c+d))=c+d  =S(a)+S(b)  and for all λ∈BR  S(λa)=S(λT(c))=S(T(λc))=λc=λS(a)  Your proof seems good as well.

Sean Becker · Accepted Answer

Step 1  I come up with a different proof as below.  When Ax=0,  𝕟x=1Rn(x)=(S∘T)(x)=S(Ax)=S(0)=S(A0)  𝕟=1Rn(0)=0  By Inverse Theorem, A is invertible, and the system Ax=y has at least one solution x for every choice of column y. Now for any 𝕟y∈Rn  S(y)=S(Ax)=x=A−1Ax=A−1y  This shows that S=TA−1

How do we know that S must be induced by some matrix B? Functions T:\mathbb{R}^n\to \mathbb{R

Answered question

Answer & Explanation

New Questions in Linear algebra