Finding a Basis for the Nullspace of a 2x2 Matrix Transformation

[charlies1902]

The problem is attached.

I am instructed to find a basis for the nullspace of T.A basis for a 2x2 matrix is
1 0
0 0

0 1
0 0

0 0
1 0

0 0
0 1Applying the transformation to each of these gives
0 0
0 0

0 2
0 0

0 0
-2 0

0 0
0 0
respectively.

Now this is where I get stuck. How do I find a basis after knowing these 4 matrices?

It look like it's a linear combo of 2 matrices:
0 1
0 0
and
0 0
1 0

but the answer in the book is
1 0
0 0

0 0
0 1

I don't undertand.

 

[Mark44]

The problem is attached.

I am instructed to find a basis for the nullspace of T.A basis for a 2x2 matrix is
1 0
0 0

0 1
0 0

0 0
1 0

0 0
0 1Applying the transformation to each of these gives
0 0
0 0

0 2
0 0

0 0
-2 0

0 0
0 0
respectively.

Now this is where I get stuck. How do I find a basis after knowing these 4 matrices?

It look like it's a linear combo of 2 matrices:
0 1
0 0
and
0 0
1 0

but the answer in the book is
1 0
0 0

0 0
0 1

I don't undertand.

Which matrices get sent to (transformed to) the zero matrix? That's the basis for Null(T).

 

[charlies1902]

Gotcha, I keep it getting it cnofused with range (T), nullspace(T) is solved by setting the right hand side of the transformation equal to 0 right?So for this case,
0 1
0 0
and
0 0
1 0

would be a basis for the range of T?

 

[Mark44]

I think so, but I didn't check your work.