Matrix Representation of a Linear Map¶

Definition¶

Let $V$ and $W$ be finite-dimensional vector spaces over the same field, with an ordered basis $\mathcal{B}=(\boldsymbol{v}_1,\ldots,\boldsymbol{v}_n)$ of $V$ and an ordered basis $\mathcal{C}=(\boldsymbol{w}_1,\ldots,\boldsymbol{w}_m)$ of $W$. For a linear map $A\colon V\to W$$W$, write each image of a basis vector in the basis $\mathcal{C}$:

The matrix representation of $A$ with respect to $\mathcal{B}$ and $\mathcal{C}$ is the $m\times n$ matrix $[A]_{\mathcal{C},\mathcal{B}}=(a_{ij})$$(a_{ij})$. Its $j$-th column is the coordinate vector $[A\boldsymbol{v}_j]_{\mathcal{C}}$. When $V=W$ and $\mathcal{B}=\mathcal{C}$$\mathcal{C}$, one writes $[A]_{\mathcal{B}}$.

Interpretation¶

A linear map is fixed by its values on a basis, so the $n$ columns $[A\boldsymbol{v}_j]_{\mathcal{C}}$ carry all the information about $A$. The matrix turns the abstract map into arithmetic on coordinates: $[A\boldsymbol{v}]_{\mathcal{C}}=[A]_{\mathcal{C},\mathcal{B}}\,[\boldsymbol{v}]_{\mathcal{B}}$ for every $\boldsymbol{v}\in V$$V$.

The figure uses the differentiation example below, with $\mathcal{B}=(1,t,t^2)$ and $\mathcal{C}=(1,t)$$(1,t)$. Going right along the top applies the map $D$ itself; going down takes coordinates. Both paths from $p$ to $(b,2c)^{\top}$ agree: differentiating and then taking coordinates gives the same vector as taking coordinates and then multiplying by $[D]$.

The matrix depends on the map and on both chosen bases, including their order. A frequent mistake is to put the coordinates of $A\boldsymbol{v}_j$ in rows instead of columns, or to write them in the standard basis when $\mathcal{C}$ is a different basis. For a map $\mathbb{R}^n\to\mathbb{R}^m$ with the standard bases the representation is the standard matrix from Math II.

The idea matters for eigenvalue problems, which are basis-free questions about a map: choosing a basis of eigenvectors makes the representing matrix diagonal, which is the content of diagonalization.

Key Results¶

• For fixed bases, $A\mapsto[A]_{\mathcal{C},\mathcal{B}}$ is a bijective linear map from $\mathcal{L}(V,W)$ onto the $m\times n$ matrices; hence $\dim\mathcal{L}(V,W)=mn$$mn$.
• Composition becomes matrix multiplication: for $B\colon U\to V$ and basis $\mathcal{D}$ of $U$, $[AB]_{\mathcal{C},\mathcal{D}}=[A]_{\mathcal{C},\mathcal{B}}[B]_{\mathcal{B},\mathcal{D}}$ (see product of linear maps).
• For the identity map on $V$ and two ordered bases $\mathcal{B},\mathcal{C}$ of $V$, $[I]_{\mathcal{C},\mathcal{B}}$ equals the identity matrix $I_n$ if and only if $\mathcal{C}=\mathcal{B}$$\mathcal{B}$; otherwise it converts $\mathcal{B}$-coordinates into $\mathcal{C}$-coordinates (see change of basis).
• If $P$ is the invertible matrix whose columns are the $\mathcal{B}$-coordinates of a second basis $\mathcal{B}'$ of $V$, then $[A]_{\mathcal{B}'}=P^{-1}[A]_{\mathcal{B}}P$$P^{-1}[A{]}_{\mathcal{B}}P$. Representations of one operator are therefore similar and share determinant, trace and eigenvalues.
• An operator on $V$ is invertible if and only if one (equivalently every) representing matrix is invertible.

Examples¶

A map $\mathbb{R}^2\to\mathbb{R}^3$${\mathbb{R}}^3$. For $A(x,y)=(x,\,y,\,x+y)$ with standard bases, $A\boldsymbol{e}_1=(1,0,1)^{\top}$ and $A\boldsymbol{e}_2=(0,1,1)^{\top}$$(0,1,1{)}^{\top }$, so

Differentiation. Let $D\colon P_2\to P_1$$P_1$, $p\mapsto p'$$p^{\prime }$, with bases $(1,t,t^2)$ and $(1,t)$. Then $D1=0$$0$, $Dt=1$$1$, $Dt^2=2t$$2t$, so $[D]=\begin{pmatrix}0&1&0\\0&0&2\end{pmatrix}$$\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 2\end{array}\right)$. Check: $p=a+bt+ct^2$ has coordinates $(a,b,c)^{\top}$ and $[D]\,(a,b,c)^{\top}=(b,2c)^{\top}$$(b,2c{)}^{\top }$, i.e. $p'=b+2ct$$2ct$.

Same operator, two bases. The operator with standard matrix $A=\begin{pmatrix}2&1\\1&2\end{pmatrix}$ satisfies $A(1,1)^{\top}=(3,3)^{\top}$ and $A(1,-1)^{\top}=(1,-1)^{\top}$$(1,-1{)}^{\top }$. In the basis $\mathcal{B}'=\big((1,1)^{\top},(1,-1)^{\top}\big)$ its representation is $\begin{pmatrix}3&0\\0&1\end{pmatrix}$: same map, same eigenvalues $3$ and $1$, simpler matrix.

Prerequisites¶

• Linear Map
• Basis
• Coordinate
• Matrix Product

Related Concepts¶

• Change of Basis
• Similarity Transformation
• Complex Conjugate Pair of Eigenvalues
• Diagonalization

Sources¶

• Math V: Advanced Linear Algebra · Week 1

Aliases: matrix of a linear map, matrix of a linear transformation, representing matrix Keywords: [A]_{C,B}, standard matrix, coordinate vector, columns are images of basis vectors Symbols: element of, real numbers, sum

Last updated: 14 October 2026 08:21