Finding a basis for the Kernel of T

[mmcgirr4]

Homework Statement

So the question is a map T: R^2x2 ---> R^2x2 by T(A) = BAB, where B = (1 1)
(1 1)

so i made A = (a c) and T(A) = ((a+b) + (c+d) (a+b) + (c+d))
(b d) ((a+b) + (c+d) (a+b) + (c+d))

now it asks Find a basis for the kernel of T and compute the dimension of the kernel T.

Homework Equations

The Attempt at a Solution

This is what I have, but I am not quite sure its right.

ker(T) = {VεR^2x2 : T (V) = 0(vector) R^2x2}

= {(a c) : [(a+b) + (c+d) (a+b) + (c+d)] = [0 0] }
{(b d) [a+b) + (c+d) (a+b) + (c+d)] [0 0] }

then

a = -b -c -d
b = -a-c-d
c = -a-b-d
d = -a-b-c

therefor the Ker(T) = {(a c) : a, b, c, d ε R^2x2}
{(b d) }

 

[Mark44]

Homework Statement

So the question is a map T: R^2x2 ---> R^2x2 by T(A) = BAB, where B = (1 1)
(1 1)

so i made A = (a c) and T(A) = ((a+b) + (c+d) (a+b) + (c+d))
(b d) ((a+b) + (c+d) (a+b) + (c+d))

now it asks Find a basis for the kernel of T and compute the dimension of the kernel T.

Homework Equations

The Attempt at a Solution

This is what I have, but I am not quite sure its right.

ker(T) = {VεR^2x2 : T (V) = 0(vector) R^2x2}

= {(a c) : [(a+b) + (c+d) (a+b) + (c+d)] = [0 0] }
{(b d) [a+b) + (c+d) (a+b) + (c+d)] [0 0] }

Isn't this the same as saying all of the 2 x 2 matrices whose entries add to 0?

then

a = -b -c -d
b = -a-c-d
c = -a-b-d
d = -a-b-c

There's a more systematic way to do this.

a = -b -c -d
b = b (obviously)
c = ... c (ditto)
d = ... d (ditto)

If you stare at the right side above, you should be able to see three vectors, many of whose entries are zero.

therefor the Ker(T) = {(a c) : a, b, c, d ε R^2x2}
{(b d) }