Letuandvbe vectors in R^{n}, and letTbe a linear operator on R^{n}. Prove that T(n) * T(v) =n*vif and only if A^{T}= A^{-1}where A is the standard matrix forT.

Any help would be great.

Printable View

- December 30th 2012, 05:52 PMrocker71Linear Algebra Proof
Let

**u**and**v**be vectors in R^{n}, and let*T*be a linear operator on R^{n}. Prove that T(**n**) * T(**v**) =**n*****v**if and only if A^{T}= A^{-1}where A is the standard matrix for*T*.

Any help would be great. - December 30th 2012, 07:51 PMzhandeleRe: Linear Algebra Proof
What does n mean? Does it mean a vector normal to v?

The transpose equaling the inverse means that the columns of A form an orthonormal basis. - December 30th 2012, 08:50 PMDevenoRe: Linear Algebra Proof
i think it's a typo. i believe the OP is asking to show that every inner-product preserving linear operator has an orthogonal matrix in the standard orthonormal basis, and that an orthogonal matrix represents an inner-product preserving linear operator.

the whole of this proof rests on the fact that (for the standard inner product in R^{n}):

so if:

for all**u**,**v**, then:

that is:

for all**u**, so

so that:

for all**v**, thus:

these steps are all reversible.