# Thread: dimention of kernel

1. ## dimention of kernel

A is a 2X2 matrices.T is a transformation from $\displaystyle M_{2X2}->M_{2X2}$

T differs zero. T is not isomorphism
prove that dim ImT=dim KerT=2

if T differs zero then its representative matrices A differs zero too.
because T is not isomorphism then its singular so we have only one row
of seros in A.

so rank(A)=1 and null(A)=1
so there is only one k 2X2 matrices which sends for which T(k)=0

the solution in my book says that this k matrices is a tranposed
vector (a,b)
??

why??
our result has to be 2x2 matrices
whith
transposed (a,b) we cant get 2x2 matrices

?

2. ## Re: dimention of kernel

Originally Posted by transgalactic
A is a 2X2 matrices. T is a transformation from $\displaystyle M_{2X2}->M_{2X2}$ T differs zero. T is not isomorphism prove that dim ImT=dim KerT=2
This is nonsense. What has to do $\displaystyle A$ in respect to $\displaystyle T$ ?

3. ## Re: dimention of kernel

if T maps a 2x2 matrix to a 2x2 matrix, then a representative matrix for T is 4x4. it is not clear from your question what "A" is.