On the figure in the plane, a standard basis $e=(i,j)$ is given and another basis $a=(a_1,a_2).$
1) A vector $u$ has the coordinates $(5,-1)$ with respect to the basis $e$. Determine the $a$-coordinates of $u$.
2) A vector $v$ has the coordinates $(-1,-2)$ with respect to basis $a$. Determine the $e$-coordinates of $v$.
This is a new subject for me, and I'm having trouble understanding the concept. How am I meant to approach this?
From the graph you can see that
$$ a_1 = i - 2j ~~~\mbox{and}~~~ a_2 = i + j $$
which can be solved for $i$ and $j$:
$$ i = \frac{1}{3}a_1 + \frac{2}{3}a_2 ~~~\mbox{and}~~~ j = -\frac{1}{3}a_1 + \frac{1}{3}a_2 $$
$(5,-1)_e$
\begin{eqnarray} u = (5,-1)_e &=& 5 i - j = \frac{1}{3}(5(a_1 + 2a_2) - (-a_1 + a_2)) \\ &=& \frac{1}{3}(6a_1 + 9a_2) = 2a_1 + 3a_2 = (2, 3)_a \end{eqnarray}
$(-1,-2)_a$
\begin{eqnarray} v = (-1,-2)_a &=& -a_1 -2a_2 = -(i - 2j) - 2(i + j) = -3i \\ &=& (-3,0)_e \end{eqnarray}
EDIT: The figure shows the representation of the vector $u$ (orange) in both basis. It can be seen that indeed $(5,-1)_e = (2, 3)_a$