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Let T: Rn → Rn be an invertible linear transformation, and let S and U be functions from Rn into Rn such that

S(T(x)) = x and U (T(x)) = x for all x in R^n
Show that
U(v)= S(v) for all v in R^n

Required:
Compute S(v) and U(v).

Answer :

Answer:

Follows are the solution to this question:

Step-by-step explanation:

[tex]T: 1 \ R^n \to 1 \ R^n[/tex] is invertible lines transformation

[tex]S[T(x)]=x \ and \ V[T(x)]=x \\\\t'x \ \varepsilon\ 1 R^n\\\\[/tex]

T is invertiable linear transformation means that is  

[tex]T(x) =A x \\\\ where \\\\ A= n \times n \ \ matrix[/tex]

and [tex]\ det(A) \neq 0 \ \ that \ is \ \ A^{-1} \ \ exists[/tex]

Let

[tex]V \varepsilon\ a\ R^{n} \ consider \ \ u= A^{-1} v \varepsilon 1 R^n\\\\T(u)= A(A^{-1} v)=(A \ A^{-1}) \\\\ v= I_{n \times n} \cdot v = v[/tex]

so,  

[tex]s[T(u)]=v[T(u)]\\\\s(v)=v(v) \ \ \forall \ \ v \ \ \varepsilon \ \ 1 R^n[/tex]

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