# Basis Change Matrices

Change of basis matrices but I'm having trouble with this question due to its context.

Let $V$ be the vector space of polynomials of degree at most $2$ over $\mathbb{R}$. Let $\mathbf{e}_1$, $\mathbf{e}_2$, $\mathbf{e}_3$ and $\mathbf{e}'_1$,$\mathbf{e}'_2$,$\mathbf{e}'_3$ be the bases $1$, $x$, $x^2$ and $1$, $(1-x)$, $(1+x)^2$ of $V$.

Write down the change of basis matrices from basis $\mathbf{e}'_i$ to $\mathbf{e}_i$, and from $\mathbf{e}_i$ to $\mathbf{e}'_i$.

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To change from the $e_i'$ basis to the $e_i$ basis, just expand each $e_i'$ in terms of the $e_i$ basis, i.e., $e_i'=a_{1i}e_1+\cdots +a_{3i}e_3$, then the $i$th column of the change of basis matrix will be $(a_{1i},a_{2i},a_{3i})^T$.