Linear Algebra proof with Linear Transformations

[CDrappi]

Homework Statement

Suppose that A is a real symmetric n × n matrix. Show that if V is
a subspace of R^n and that A(V) is contained in V , then A(V perp) is contained in V perp.

Homework Equations

A = A_T (A is equal to its transpose)

The Attempt at a Solution

I have no idea where to start

 

[micromass]

Take a vector v\in V^\bot (what does this mean??).
You'll need to show that Av\in V^\bot (what do you need to show for that?)

 

[CDrappi]

I still am not sure what to do. Any further helpings?

 

[micromass]

What does v\in V^\bot mean??
What does Av\in V^\bot mean??
Just give the definition...

 

[CDrappi]

What does v\in V^\bot mean??
What does Av\in V^\bot mean??
Just give the definition...

That a vector v is contained in V perp

That a vector Av is contained in V perp

 

[micromass]

Yes, of course. But what does it mean that v is contained in V^\perp. What property must hold?

 

[CDrappi]

If V = C(B), the column space of some matrix B, then Bv = 0

 

[micromass]

Hmm, how did you define V^\bot? I remember that it had to do with inner products...

 

[CDrappi]

Hmm, how did you define V^\bot? I remember that it had to do with inner products...

Wouldn't V^\bot just be the left nullspace of B?

 

[micromass]

Yes, but if you're given a set V^\bot. How do you find the matrix B??

 

[CDrappi]

Yes, but if you're given a set V^\bot. How do you find the matrix B??

I do not know. Help prease!

 

[micromass]

Given a subspace V. How did your course define the subspace V^\bot??

 

[Gear300]

For a subspace V spanned by the column space of a matrix V, Ker(V^T) returns V^\bot (orthogonal complement to V). If v is in V and v_p is in the orthogonal complement, then v_p is in Ker(V^T). Av_p should also be in Ker(V^T). If it is, then the inner product of Av_p and v should be 0.

 

[CDrappi]

Oh. We defined it as whatever part of R^n that V isn't in

 

[CDrappi]

For a subspace V spanned by the column space of a matrix V, the Ker(V^T) returns V^\bot.

Can you write the last part of that out to make it a little clearer? I can't understand exactly what you mean.

 

[CDrappi]

hmph. it seems you've edited it on me