I'm a little confused by this. I've been watching some videos on change of bases, but they tend to talk about the change of basis matrix.
Let's say we have coefficients of a vector V in base $B = (e1,e2,e3)$. To get the coefficients of $V$ in the canonical basis, we do the following
$[e1 e2] * [V]_B = [V]_C$ (where $C$ is the canonical basis). Is this right so far?
Ok, so this is the online videos I've found have explained how to change a basis. But how do I do it if I want to convert my vector so that it's in another canonical base? Would I just have to turn it into the standard base, and then back into a different base, or is there an easier method? Thanks.
You express the coordinates of the target basis into the original basis and then it's the same method that you use for the canonical basis.
For instance if you have a basis $\mathcal{B_1} = (e_1,e_2,e_3)$ and a basis $\mathcal{B_2} = (f_1,f_2,f_3)$ and let's say $V_1$ is the column representation of a vector in $\mathcal{B_1}$ and $V_2$ the same vector in $\mathcal{B_2}$.
You have a certain relation between the basis. Namely, there exists 9 scalars such that: $$\begin{cases} f_1 = \alpha_1e_1 + \alpha_2e_2 + \alpha_3e_3 \\ f_2 = \beta_1e_1 + \beta_2e_2 + \beta_3e_3 \\ f_3 = \gamma_1e_1 + \gamma_2e_2 + \gamma_3e_3 \end{cases}$$
Then the change of basis matrix is: $$ P = \begin{pmatrix} \alpha_1 & \beta_1 & \gamma_1 \\ \alpha_2 & \beta_2 & \gamma_2 \\ \alpha_3 & \beta_3 & \gamma_3 \end{pmatrix}$$
And then $V_2 = P^{-1}V_1$