# Finding Linear Transformation

• Apr 6th 2013, 08:34 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, 10:03 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, 10:17 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 \$\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).\$