TITLE: Find coordinate mapping of $V =F^2,B= \langle e_1,e_1+e_2\rangle$ Linear Algebra
QUESTION [2 upvotes]: Describe the coordinate mappings for the given bases on the given
vector spaces B
$V =F^2,B= \langle e_1,e_1+e_2\rangle$ and
$V = P_{\le n}(F), B = \langle 1,x,2x^2,\ldots,nx^n\rangle$

I don't know what the question is actually asking. What does it mean to be a coordinate map?

REPLY [1 votes]: A coordinate map is an identification between linear combinations and scalar vectors with respect to an ordered basis. E.g. $2e_1 + e_2 - e_4 \leftrightarrow (2,1,0,-1)$.  
Lets see what happens in the first case. Assume that $F^2 = \langle e_1, e_2\rangle$. Then you have $B = \langle e_1, e_1 + e_2\rangle$. You are asked for a map that satisfies $e_1 \mapsto e_1$, $e_1 + e_2 \mapsto e_2$. This map is simply a $2\times 2$ matrix that satisfies those two conditions. Deduce that the matrix is 
$$\begin{bmatrix} 1 & -1 \\ 0 & 1 \end{bmatrix}.$$
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