Finding Linear Transformation

Hi,

I'm having trouble with this subject and would like to ask for some help and explanations.

For example, I have the following question:

V1 = (1,1,1)

V2 = (2,3,1)

Find linear transformation T:R3 -> R3 such that:

a. Its kernel spans by V1 and V2

b. Its kernel span by V1 only

How should I approach this?

Any help will be appreciated.

Thanks

Re: Finding Linear Transformation

Hey MathStudent111111.

Hint: If it only spans V1 then it will only have scalar multiples of V1.

Re: Finding Linear Transformation

Just recall that a linear transformation is completely determined if we know the image of the vectors in any base of the domain.

So, first of all, find a third vector so that is a base for

A. You want the kernel of to be the span of therefore you need to set and Additionally, you have to make non-zero (this way you don't get a bigger kernel).

B. Set and send and to two non-zero linearly independent vectors in say, and