Question

#7
https://drive.google.com/file/d/0B7ss_Y8M_VEgZWpwRnA1eU9ucUE/view?usp=drivesdk

Answer 1

The transformation matrix with the given basis should be

0 0 2 0
0 0 0 6
0 0 0 0
0 0 0 0

Answer 2

I might use some help with interpreting the nullspace(kernel) and columnspace(range) of above matrix :)

Answer 3

It seems both nullspace and columnspace are the collection of all linear polynomials :

c1 + c2t

Answer 4

Because the second derivative of any linear polynomial is 0

And the second derivative of any cubic polynomial is a linear polynomial

Answer 5

So linear polynomials goto 0,
And everything goes to linear polynomials

Answer 6

Looks I'm done
Thanks