Question

Let pn denote the set of all polynomials of degree at most n with real coefficients (i.e. polynomials of the form a0+a1t+a2t 2+...+ant n for a0, a1, ..., an r) and let s = {t 2 , 1+t, t3+4}.
a.is the set s linearly independent in p3?

Answer 1

We can consider each unique $t^i$ as the as the $i$-th unit vector. So your set $s$ can be considered as the vectors 

$\langle 1, 0, 0 \rangle, \langle 0, 1, 1 \rangle,\langle 0, 3, 4 \rangle$

Then check for independence your favorite way. In this case, I'll see if the linear map A of the new basis vectors doesn't map to a subspace via the determinant not being zero:


$det(A) = det \left ( \left [ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 3 \\ 0 & 1 & 4 \end{array}\right ] \right ) = 1(4-3) = 1$

So they are linear independent.