# Finding Linear Transformation

• Apr 6th 2013, 09:34 PM
MathStudent111111
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
• Apr 6th 2013, 11:03 PM
chiro
Re: Finding Linear Transformation
Hey MathStudent111111.

Hint: If it only spans V1 then it will only have scalar multiples of V1.
• Apr 6th 2013, 11:17 PM
MATHNEM
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 $v_3$ so that $\{v_1,v_2,v_3\}$ is a base for $\Bbb R^3.$

A. You want the kernel of $T$ to be the span of $\{v_1,v_2\},$ therefore you need to set $T(v_1)=(0,0,0)$ and $T(v_2)=(0,0,0).$ Additionally, you have to make $T(v_3)$ non-zero (this way you don't get a bigger kernel).

B. Set $T(v_1)=(0,0,0),$ and send $v_2$ and $v_3$ to two non-zero linearly independent vectors in $\Bbb R^3,$ say, $(1,0,0)$ and $(0,1,0).$