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

?