# dimention of kernel

• Jun 29th 2011, 05:50 AM
transgalactic
dimention of kernel
A is a 2X2 matrices.T is a transformation from $M_{2X2}->M_{2X2}$

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

?
• Jun 29th 2011, 08:18 AM
FernandoRevilla
Re: dimention of kernel
Quote:

Originally Posted by transgalactic
A is a 2X2 matrices. T is a transformation from $M_{2X2}->M_{2X2}$ T differs zero. T is not isomorphism prove that dim ImT=dim KerT=2

This is nonsense. What has to do $A$ in respect to $T$ ?
• Jun 29th 2011, 09:33 AM
Deveno
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.