# generate

• Nov 16th 2010, 07:08 PM
mathbeginner
generate
Quote:

Let T: V-> W be a linear map, and let x1,....,xm and y1,...yn be two lists of vector in V. Suppose that

(a) x1,...xm generate Ker T and
(b) T(y1),...,T(yn) genterate W.

Show that the list x1,....,xm, y1,...,yn generates V.
Pls help
• Nov 16th 2010, 07:26 PM
Drexel28
Quote:

Originally Posted by mathbeginner
Pls help

I helped you with the other one, what have you done for this one?
• Nov 16th 2010, 07:52 PM
mathbeginner
I know I can use b and show the T is subjective.
then since T is subjective and T(y1),...(T(yn) generates W then y1,...,yn generates V.
but I don't know how to show T is subjective.

and I don't know how to start a yet.
• Nov 16th 2010, 08:09 PM
Drexel28
Quote:

Originally Posted by mathbeginner
I know I can use b and show the T is subjective.
then since T is subjective and T(y1),...(T(yn) generates W then y1,...,yn generates V.
but I don't know how to show T is subjective.

and I don't know how to start a yet.

Firstly, it's [i]surjective[/b], secondly you think that $\displaystyle \text{span}\{T(v_1),\cdots,T(v_k)\}=W\implies \text{span}\{v_1,\cdots,v_k\}=V$?
• Nov 16th 2010, 08:20 PM
mathbeginner
Quote:

Originally Posted by Drexel28
Firstly, it's [i]surjective[/b], secondly you think that $\displaystyle \text{span}\{T(v_1),\cdots,T(v_k)\}=W\implies \text{span}\{v_1,\cdots,v_k\}=V$?

but how can I show that T is surjective?
• Nov 17th 2010, 10:51 AM
manygrams
I'm working on this question as well. I thought the same but realized that it only proves some vectors in W can be generated by T(b_1*y_1 + ... + b_n*y_n). this is correct, right?

Do you have any hints? I'm pretty stuck but I've been working on it for a while.

EDIT> I was thinking of using dimV=dimT(v)+dim(KerT) (v in V), would this help at all?