Help with linear transformation

Hi guys.

I need to prove the following:

Let be a vector space over , such that .

Let be a linear transformation such that: .

Prove that for every : are linear independent.

This is what I tried so far:

Let .

suppose , I need to prove .

since is linear, and , then also , and then:

now since :

but now I'm not sure what conclusion should I draw from this...

I can really use some guidance here.

Thanks in advanced!

Re: Help with linear transformation

Quote:

Originally Posted by

**Stormey** Hi guys.

I need to prove the following:

Let

be a vector space over

, such that

.

Let

be a linear transformation such that:

.

Prove that for every

:

are linear independent.

This is what I tried so far:

Let

.

suppose

, I need to prove

.

since

is linear, and

, then also

, and then:

now since

:

but now I'm not sure what conclusion should I draw from this...

I can really use some guidance here.

Thanks in advanced!

Hi Stormey! :)

Suppose are linearly dependent.

Then there is some such that .

What do you get for ?

Re: Help with linear transformation

Hi ILikeSerena!

Thanks for the help.

let me see if I got it right:

if , then , and then we'll get a contradiction since:

(because according to the assumption )

no solution in

?

Re: Help with linear transformation