Finding Coordinate Vector?

Question

In a new basis of $\mathbb R^2$ , the coordinate vector of the vector $[2,3]$ is $[4,3]$ and that of the vector $[4,5]$ is $[6,6]$. Given this information, what is the coordinate vector of $[6,7]$?

Answer 1

It is $-[4,3]+2[6,6]=[8,9]$ because $[6,7]=-[2,3]+2[4,5]$.

Answer 2

The most efficient method is to express $[6,7]$ as a linear combination of $[2,3]$ and $[4,5]$ and then use the fact that a change of basis is a linear transformation in co-ordinates to find the new co-ordinates of $[6,7]$.

If you want to explicitly find the new basis vectors, you can proceed as follows. Suppose the new basis vectors are $\mathbf{e_1}$ and $\mathbf{e_2}$. Then

$4\mathbf{e_1}+3\mathbf{e_2}=[2,3]$

$6\mathbf{e_1}+6\mathbf{e_2}=[4,5]$

$\Rightarrow 2\mathbf{e_1} = 2[2,3] - [4,5] = [0,1]$

$\Rightarrow \mathbf{e_1} = \left[0, \frac 12 \right], \space \mathbf{e_2} = \left[\frac 23, \frac 13 \right]$

$ \Rightarrow [6,7] = 8\mathbf{e_1} + 9\mathbf{e_2}$