2. Coordinates and change of basis

Unless otherwise stated, $F$ is a field and every space in a dimension formula is finite-dimensional.

Coordinates describe a vector relative to an ordered basis. They change when the basis changes, while the vector and the linear map do not.

Definition

Matrix of a map

For ordered bases $B=(v_1,\ldots,v_n)$ of $V$ and $C=(w_1,\ldots,w_m)$ of $W$, the matrix $[T]_{C\leftarrow B}$ has column $j$ equal to $[T(v_j)]_C$. Thus $[T(v)]_C=[T]_{C\leftarrow B}[v]_B$. For $S:W\to X$ and an ordered basis $D$ of $X$, $[ST]_{D\leftarrow B}=[S]_{D\leftarrow C}[T]_{C\leftarrow B}$.

A basis $B=(b_1,\ldots,b_n)$ identifies $V$ with $F^n$ through its coordinate isomorphism. For bases $B,C$, the matrix $[T]_{C\leftarrow B}$ has columns $[Tb_j]_C$. For an endomorphism, changing basis by $[v]_B=P[v]_{B'}$ gives

$$[T]_{B'}=P^{-1}[T]_BP.$$

This is similarity, with the same basis change in domain and codomain. Independent changes of domain and codomain for a general map produce $Q^{-1}AP$, a different equivalence problem.

Proof

Follow the coordinates

For every $v$, the old output coordinates are $AP[v]_{B'}$. They also equal $Q[T(v)]_{C'}$. Since the basis-change matrices are invertible, solving for the new output coordinates gives $Q^{-1}AP$. In the endomorphism case $Q=P$.

Worked example

A nonstandard polynomial basis

On the real polynomials $P_1$ of degree at most one, let $T(p)=p'$ and $B'=(1+x,1)$. The derivative of $1+x$ is $1=0(1+x)+1(1)$, and the derivative of $1$ is zero. Therefore $[T]_{B'\leftarrow B'}=\begin{pmatrix}0&0\\1&0\end{pmatrix}$. Relative to $B=(1,x)$ the matrix is $A=\begin{pmatrix}0&1\\0&0\end{pmatrix}$; $P=\begin{pmatrix}1&1\\1&0\end{pmatrix}$ and indeed $P^{-1}AP$ is the stated matrix.

Similarity preserves characteristic polynomial, determinant, trace, rank and eigenvalues, since it represents the same endomorphism in another basis. Equal determinant and trace alone do not imply similarity. Row operations need not be similarity transformations; they generally change the represented endomorphism. To choose the direction of a change matrix, write the coordinate equation before multiplying.

Exercises

1. Let $T(x,y)=(x+2y,y)$ on $\mathbb R^2$ and $B'=((1,0),(1,1))$. Find $[T]_{B'\leftarrow B'}$.
2. Show similar matrices have the same characteristic polynomial, using $\chi_A(t)=\det(tI-A)$.
3. Are $I_2$ and $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ similar? Justify.

Full solutions

1. $T(1,0)=(1,0)$ has coordinates $(1,0)$ in $B'$. $T(1,1)=(3,1)=2(1,0)+(1,1)$ has coordinates $(2,1)$. The matrix is $\begin{pmatrix}1&2\\0&1\end{pmatrix}$.
2. If $B=P^{-1}AP$, then $tI-B=P^{-1}(tI-A)P$. Multiplicativity of determinant gives $\det(tI-B)=\det(P^{-1})\det(tI-A)\det(P)=\det(tI-A)$.
3. No. For every invertible $P$, $P^{-1}I_2P=I_2$, which is not the second matrix. Their equal characteristic polynomials do not suffice.

Connection

An eigenbasis is a basis in which an endomorphism has diagonal matrix. The next unit asks when such a basis exists.