Change of Basis

If $\mathbb{V}$ is a vector space with basis $B$ and $\mathbb{W}$ is a basis with basis $C$, and $T:\mathbb{V}\to \mathbb{W}$ is linear, then there exists a matrix $A_B^C$ such that $(Tv)_C=A_B^C(v)_B$. The $j$th column consists of the coordinate representation of the $j$th vector in $\mathbb{V}$'s basis expressed with respect to basis $C$. This is just a more general form of how we've been talking about transformations: before, we weren't including bases, but now we're explicitly stating that the transformation is from one basis to another. If $B$ and $C$ are two bases of vector space $\mathbb{V}$, then there exists a transition matrix $P_B^C$ (read: "$P$ from $B$ to $C$") such that $(v)_C=P_B^C (v)_B$ for all $v\in \mathbb{V}$.

If $\mathbb{V}$ is a vector space with bases $B$ and $C$ and $T:\mathbb{V}\to \mathbb{V}$ is linear, then $P_B^CA_B^B=A_C^CP_B^C$.

So how do we change basis?
We can change basis by multiplying by the transition matrix! If we have a vector $v$ in basis $B$ (denoted "$(v)_B$"), we can find its representation in basis $C$ by multiplying by $P_B^C$. So, $(v)_C=P_B^C(v)_B$. That's the same thing as a 2d transformation! We use the original coordinate system, and transform the basis vectors to the other system's coordinates. So, changing basis vectors is matrix-vector multiplication, where the matrices' columns are how we would express the "other" basis vectors!

Change of Basis for Transformations
We can change the basis that a transformation is represented in the same way we would change basis for vectors: $$A_C^C=P_B^CA_B^BP_B^C.$$ If you read this out loud, it'll make sense: "$A$ from base $C$ to $C$ is equal to the transition matrix from $C$ to $B$ times $A$ from $B$ to $B$ times the transition matrix from $B$ to $C$." We'll see an incredibly powerful application of this in the Diagonalization section.