Question

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 1

Follows are the solution to this question:

Explanation:

`T: 1 \ R^n \to 1 \ R^n` is invertible lines transformation

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

T is invertiable linear transformation means that is

`T(x) =A x \\\\ where \\\\ A= n * n \ \ matrix`

and
`\ det(A) \\eq 0 \ \ that \ is \ \ A^(-1) \ \ exists`

Let

`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 * n) \cdot v = v`

so,

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