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 $\displaystyle v_3$ so that $\displaystyle \{v_1,v_2,v_3\}$ is a base for $\displaystyle \Bbb R^3.$

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

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