# Change of basis

Hi everyone,

I am working on the following problem.

Suppose the set of vectors X1,..,Xk is a basis for linear space V1.
Suppose the set of vectors Y1,..,Yk is also a basis for linear space
V1.
Clearly the linear space spanned by the Xs equals the linear space
spanned by the Ys.

Set
X=[X1: X2 :...: Xk]
Y=[Y1: Y2 :...: Yk]

Construct an algebraic argument to show that
X(X'X)^(-1)X'=Y(Y'Y)^(-1)Y'

This is the idea I have:
X=PYP^{-1}
where P changes the basis from Y to X.

Is this the right avenue?Thanks in advance.

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HallsofIvy
Homework Helper
Didn't we just have this? Use the fact that (AB)-1= B-1A-1.

Thanks for the help.

There is a problem with the dimensions.

X is n*k

because each of X1,..,Xk is n*1

So I was going to write (using your help)
X(X'X)^{-1}X'= X(X^{-1}X'^{-1})X'
But X^{-1} does not exist

Last edited:
I think I need to use the fact that X(X'X)^(-1)X' is a projection operator somehow.

HallsofIvy
Homework Helper
If all columns in X are linearly independent- and you said they formed a basis for V1- then X must be non-singular and have an inverse!

X is not a square matrix

HallsofIvy