Comparing Basis Vectors in Linear Spaces: X and Y

[julie94]

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.

 

[HallsofIvy]

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

 

[julie94]

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

 

[julie94]

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

 

[HallsofIvy]

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!

 

[julie94]

X is not a square matrix

 

[HallsofIvy]

Ah! so A and B span V1, a subspace of V!

 

[julie94]

yes

any ideas?