Question

For the linear transformation T, determine if it is invertible and justify your answer

T: P3(R) -> R^3 by T(p(x)) = (p(0), p(1), p(2))

Answer 1

So this function will be invertible if it is one-to-one.

Is it one-to-one?

Answer 2

Not quite sure. That was the whole question, no hints what so ever

Answer 3

Yes, think about it. Write down an arbitrary member of P3, call it p(x) look at T(p(x)) and ask yourself if there is another poly q(x) such that T(q(x)) = T(p(x))

Answer 4

What is P3(R) in this case?

Answer 5

Third order polynomials over the real numbers.

Answer 6

Another approach is to ask yourself what is the dim of P3(R) and R^3.

Answer 7

Indeed that would work too, as T is a vector space homomorphism.

Answer 8

Yes. In fact, it's not hard to convince yourself that T must be onto and hence by the first isomorphism theorem
\[ P_3 / \ker(T) \simeq \mathbb{R}^3 \]
and therefore ker(T) is non-zero. Hence T is not 1:1 and T cannot be invertible.

Answer 9

If that's a bit complex, then do this. Make the association between P3 and R^4 by a natural isomorphism which associates this basis of P3 {1,x,x^2,x^3} with this basis of R^4
{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) }

Then you can think of T as a linear map from R^4 to R^3
T(a,b,c,d) = (a, a+b+c+d, a+2b+4c+8d)

Now use regular matrix considerations to determine whether or not the matrix representation of T is invertible.