Hi guys.

I need to prove the following:

Let be a linear transformation.

Let be the span of a set , and be the span of a set .

I need to prove that if for every then .

I'm not sure where to start, I need some guidance please.

thanks in advanced!

Printable View

- May 23rd 2013, 08:11 AMStormeyInner product space and linear transformation proof
Hi guys.

I need to prove the following:

Let be a linear transformation.

Let be the span of a set , and be the span of a set .

I need to prove that if for every then .

I'm not sure where to start, I need some guidance please.

thanks in advanced! - May 23rd 2013, 08:05 PMchiroRe: Inner product space and linear transformation proof
Hey Stormey.

For your problem can you show that if u is non-zero (in general) then since <u,v> = 0 implies u = 0 or v = 0, then if v is not necessarily zero then u = 0.

if u = Tx and x is not necessarily zero, then you need to show T = 0 but for all x.

So the proof now becomes showing that T = 0 satisfies Tx = 0 for all valid vectors in x. You could try proof by contradiction by assuming that T is non-zero and find a value for some realization of x' where Tx = 0 but Tx' != 0 for x' != x. This will prove the statement that T must be zero in order for Tx = 0 for all valid x in the vector space. - May 24th 2013, 01:54 AMStormeyRe: Inner product space and linear transformation proof
- May 24th 2013, 03:01 AMchiroRe: Inner product space and linear transformation proof
When I say <u,v> = 0 I mean when this holds for all possible v where v can vary. I should have been more clear on that.

Also x is in set S. - May 24th 2013, 08:13 AMStormeyRe: Inner product space and linear transformation proof
OK, thanks!