# Kernel and range of linear transformation

• Apr 13th 2009, 05:30 PM
hasanbalkan
Kernel and range of linear transformation
Hi everyone,

I need some help with the following problem:

Let U be the vector space of 2 x 2 matrices. Let A and B be 2 x 2 matrices. Find the kernel and range of each linear transformation:

1) T(A) = trace(A)
2) T(A) = A + B

Any help is greatly appreciated!
• Apr 13th 2009, 06:26 PM
ziggychick
Quote:

Originally Posted by hasanbalkan
Hi everyone,

I need some help with the following problem:

Let U be the vector space of 2 x 2 matrices. Let A and B be 2 x 2 matrices. Find the kernel and range of each linear transformation:

1) T(A) = trace(A)
2) T(A) = A + B

Any help is greatly appreciated!

Remember the Kernel of T is the subset of U which T takes to zero. That is, a matrix A is in the kernel if and only if T(A)=0.

So for the first transformation any matrix A where T(A)=Tr(A)=0 will be in the kernel. So what is necessary about the diagonal entries of A if Tr(A)=0?

For the range, remember the range is the subset of the codomain of T which has a pre-image in the domain. So for the first transformation the domain of T is U and the Codomain is \$\displaystyle \mathbb{R}\$. We must find the subset of \$\displaystyle \mathbb{R}\$ that is "hit" by T. This means we must find all possible real numbers \$\displaystyle a\$ for which there exists a matrix A so Tr(A)=a. All such numbers make up the range.

Is it possible to find a matrix A so Tr(A)=a for any real number \$\displaystyle a\$? If I asked, find a matrix A so Tr(A)=5, could you find one?

You should be able to apply these ideas to the other transformation as well.
Hope this helps. :-)