Let $ Y = \{ 1 + x, 1 − x + x^2 , 1 + 3 x − x^2 \} \subseteq P^2 ( \mathbb{R} ).$

Question

Show that $Y$ does not span $P^2$ and find a basis for the span of $Y$.

I can row reduce the matrix so that the bottom row is all zeros is this enough to prove that it does not span? I find it easy to show that something does span but not to show that it does not span...

The basis I got was $\{ 1 + (1/2)x^2, x + (-1/2)x^2 \}$ is this correct?

Answer 1

Yes, your calculations are correct and the basis for the span of $Y$ is good.

It would have been very helpful if you had your matrix, and its row reduced form written down, so the reader has an easier time to see your complete proof.

Answer 2

Alternative approach:

For this particular question, we are told that it doesn't span $P^2(\mathbb{R})$ but we have to prove it.

It is easy to see that the first two polynomials are not multiple of each other, I shall try to prove that the third is a linear combination of them. By observing the coefficient of $x^2$ and the constant term. I can quickly write down that

$$2(1+x)-(1-x+x^2) = (1+3x-x^2)$$

Hence a basis is $\{ 1+x, 1-x+x^2 \}$.

To check that $\{ 1+ \frac12 x^2, x-\frac12 x^2 \}$ is a valid basis:

$$(1+ \frac12 x^2)+ (x-\frac12 x^2)=1+x$$

$$(1+ \frac12 x^2)-(x-\frac12 x^2)=1-x+x^2$$

Yup, it is correct.