a.) Find a basis for the orthogonal complement to the following subspaces of $\mathbb{R^4}$. The subspace spanned by $(1,2,-1,3)^T$,$(-2,0,1,-2)^T$,$(-1,2,0,1)^T$.
b.) Use the Legendre polynomials to find the best quadractic and cupic approxiamtion to $t^4$, based on the $L^2$ norm on [-1,1].
For a, I found the orthogonal matrix from the spanned vectors given which gave me
$K=\pmatrix{1&\frac{-7}{5}&\frac{-7}{5}\\2&\frac{6}{5}&\frac{6}{5}\\-1&\frac{2}{5}&\frac{2}{5}\\3&\frac{-1}{5}&\frac{-1}{5}}$ but how can I find the basis of the spanned vectors, becuase I am confused on what they are exactly trying to get me to find.
For b, I found the quadratic $\frac{1}{5}+ \frac{4}{7}(\frac{-1}{2}+\frac{3}{2}t^2) = \frac{-3}{35} + \frac{6}{7}t^2$ but how can I find the cubic using the same process?