Matrix Representations and Change of Basis

Representing a Linear Map as a Matrix

Every linear map $h: V \to W$ between finite-dimensional spaces can be encoded as a matrix once bases are chosen for both spaces.

Construction: fix a basis $B = \langle\vec{\beta}_1, \ldots, \vec{\beta}_n\rangle$ for $V$ and $D = \langle\vec{\delta}_1, \ldots, \vec{\delta}_m\rangle$ for $W$. The matrix representation $\mathrm{Rep}_{B,D}(h)$ is the $m \times n$ matrix whose $j$-th column is the coordinate vector of $h(\vec{\beta}_j)$ with respect to $D$:

$$\mathrm{Rep}_{B,D}(h) = \left[\, \mathrm{Rep}_D(h(\vec{\beta}_1)) \;\big|\; \cdots \;\big|\; \mathrm{Rep}_D(h(\vec{\beta}_n)) \,\right]$$

Applying the map becomes matrix-vector multiplication: if $\vec{v}$ has coordinates $\mathrm{Rep}_B(\vec{v}) = \vec{c}$, then

$$\mathrm{Rep}_D(h(\vec{v})) = \mathrm{Rep}_{B,D}(h) \cdot \vec{c}$$

The representation depends on the choice of bases. Same map, different bases, different matrix.

Matrix of a Composition

If $g: V \to W$ and $f: W \to X$ are linear maps with bases $B, D, E$ respectively, the matrix of the composition is the product of the matrices:

$$\mathrm{Rep}_{B,E}(f \circ g) = \mathrm{Rep}_{D,E}(f) \cdot \mathrm{Rep}_{B,D}(g)$$

This is why matrix multiplication is defined the way it is. The non-obvious definition of matrix multiplication $(AB)_{ik} = \sum_j A_{ij}B_{jk}$ is precisely what is required for the composition law to hold.

Change of Basis

Suppose you have two bases $B$ and $\hat{B}$ for the same space $V$. The change-of-basis matrix $\mathrm{Rep}_{B,\hat{B}}(\text{id}_V)$ converts coordinates: if $\vec{v}$ has coordinates $\vec{c}$ in basis $B$, its coordinates in $\hat{B}$ are

$$\hat{\vec{c}} = \mathrm{Rep}_{B,\hat{B}}(\text{id}_V) \cdot \vec{c}$$

Constructing this matrix: column $j$ is the coordinate expression of $\vec{\beta}_j$ (the $j$-th $B$-basis vector) in terms of $\hat{B}$.

In $\mathbb{R}^n$: if $B$ is the standard basis and $\hat{B}$ is any other basis (columns of matrix $P$), then the change-of-basis matrix from $\hat{B}$ to $B$ is $P$ itself (since the columns of $P$ are the $\hat{B}$-basis vectors expressed in standard coordinates). The inverse $P^{-1}$ converts from $B$ to $\hat{B}$.

Similarity

Two matrices $A$ and $\hat{A}$ are similar if there exists an invertible matrix $P$ such that:

$$\hat{A} = P^{-1}AP$$

Similar matrices represent the same linear transformation with respect to different bases. The matrix $P$ is the change-of-basis matrix between the two coordinate systems.

Properties preserved under similarity (invariants of the transformation, independent of basis):

• Determinant: $\det(\hat{A}) = \det(A)$
• Trace: $\text{tr}(\hat{A}) = \text{tr}(A)$
• Eigenvalues (same set of eigenvalues)
• Rank and nullity
• Characteristic polynomial

When we diagonalize a matrix — find $P^{-1}AP = \Lambda$ where $\Lambda$ is diagonal — we are finding the basis in which the transformation acts most simply: pure scaling along independent axes.

Inverse of a Map

If $h: V \to W$ is an isomorphism, its inverse $h^{-1}: W \to V$ is also a linear map (it can be checked directly from the axioms). Its matrix representation satisfies:

$$\mathrm{Rep}_{D,B}(h^{-1}) = \left[\mathrm{Rep}_{B,D}(h)\right]^{-1}$$

The matrix inverse corresponds to the inverse map. Computing the inverse of a matrix $A$ by row-reducing $[A \mid I]$ to $[I \mid A^{-1}]$ is equivalent to finding the change-of-basis from the identity to $A$.