Finding the dimension of a polynomial subspace

Question

Polynomial Subspace

For this question (in the attached link above) I originally thought the answer was 7 because typically the equation for finding the dimension of a polynomial is Pn+1 --> in this case it would be 6+1. However, since this is incorrect I was wondering if the answer would be 4 because I could only figure out the elements within W being 1, t^2, t^4, and t^6. Would this be correct?

Answer 1

I would prefer a slightly different wording, but that's the basic gist of it, yeah. The answer is $4$ because of the four polynomials $1, t^2, t^4, t^6$.

More formally, those four polynomials are linearly independent, as the only way to make $$ a\cdot 1 + b\cdot t^2 + c\cdot t^4 + d\cdot t^6 = 0 $$ is to have $a = b = c = d = 0$ (at least as long as we require that $a, b, c, d$ are real numbers, which we do). Also, any polynomial in $W$ is a linear combination of these four polynomials (easily seen by just writing down the polynomial and looking at it).

Together this means that these four polynomials form a basis for $W$, and this implies that $W$ has dimension $4$.