Chapter 34. Matrix Representation of Linear Maps

A linear map can be studied abstractly as a function between vector spaces. It can also be studied concretely as a matrix. The connection between these two descriptions is one of the main bridges in linear algebra.

If $V$ and $W$ are finite-dimensional vector spaces over the same field $F$, and

is linear, then $T$ can be represented by a matrix after bases are chosen for $V$ and $W$. The matrix stores the coordinates of the images of the basis vectors of $V$. In the standard coordinate spaces, a linear transformation $T:\mathbb{R}^n\to\mathbb{R}^m$ has an $m\times n$ matrix $A$ such that $T(x)=Ax$. The columns of $A$ are the images of the standard basis vectors.

34.1 The Basic Idea

A linear map is determined by what it does to a basis.

Let

be a basis of $V$. Every vector $v\in V$ can be written uniquely as

If $T:V\to W$ is linear, then

Thus, to know $T(v)$ for every $v$, it is enough to know

A matrix representation is a compact way to store these output vectors.

34.2 Standard Matrix Representation

Consider a linear transformation

Let

be the standard basis of $\mathbb{R}^n$. The standard matrix of $T$ is the $m\times n$ matrix

That is, the first column of $A$ is $T(e_1)$, the second column is $T(e_2)$, and so on.

Then, for every

we have

By linearity,

But this is exactly matrix multiplication:

So a linear transformation on coordinate space is the same data as a matrix.

34.3 Example in $\mathbb{R}^2$

Define

by

The standard basis vectors are

Compute their images:

and

Therefore the standard matrix of $T$ is

Indeed,

This agrees with the original definition of $T$.

34.4 Matrix Size

The size of the representing matrix is determined by the dimensions of the domain and codomain.

If

then the matrix of $T$ has $m$ rows and $n$ columns.

The number of columns equals the number of input coordinates. The number of rows equals the number of output coordinates.

Thus:

This convention matches matrix-vector multiplication:

The input vector has $n$ entries, and the output vector has $m$ entries.

34.5 From Matrix to Linear Map

Every $m\times n$ matrix defines a linear map.

Let

Define

by

Then $T_A$ is linear. For vectors $u,v\in F^n$ and scalar $c\in F$,

and

Thus matrices and linear maps between coordinate spaces are equivalent descriptions.

The matrix is the data. The linear map is the action.

34.6 From Linear Map to Matrix

Every linear map between finite-dimensional coordinate spaces gives a matrix.

Let

be linear. Let $e_1,\ldots,e_n$ be the standard basis of $F^n$. Define

Then

for every $x\in F^n$.

This construction is unique. If another matrix $B$ satisfies

for every $x$, then in particular

for each $j$. But $Be_j$ is the $j$-th column of $B$. Hence every column of $B$ equals the corresponding column of $A$. Therefore

So a linear transformation between coordinate spaces has one and only one standard matrix.

34.7 Coordinates Relative to a Basis

The previous sections used the standard basis. For abstract vector spaces, or for nonstandard coordinate systems, we must use coordinates relative to chosen bases.

Let

be an ordered basis of $V$. If

then the coordinate vector of $v$ relative to $B$ is

The order of the basis matters. Changing the order changes the coordinate vector.

For example, in $\mathbb{R}^2$, let

If

then we solve

This gives

Solving,

Therefore

34.8 Matrix of a Linear Map Relative to Bases

Let

be linear. Let

be an ordered basis of $V$, and let

be an ordered basis of $W$.

The matrix of $T$ relative to $B$ and $C$ is the $m\times n$ matrix whose $j$-th column is the coordinate vector of $T(v_j)$ in the basis $C$:

The notation $C\leftarrow B$ means that inputs are written in basis $B$, and outputs are written in basis $C$.

The defining relation is

This equation is the coordinate form of the linear map.

34.9 Example with Nonstandard Bases

Let

be defined by

Let the domain basis be

and let the codomain basis be the standard basis

Compute the images of the basis vectors:

and

Since $E$ is the standard basis, these are already their coordinate vectors in $E$. Therefore

With respect to this input basis, the transformation looks like simple scaling by $2$.

This illustrates an important point: the same linear map can have different matrices under different choices of bases.

34.10 The Same Map, Different Matrices

The matrix is not the transformation itself. It is the coordinate description of the transformation.

A linear map $T:V\to W$ exists independently of coordinates. To write it as a matrix, we choose a basis for $V$ and a basis for $W$. Different choices may produce different matrices.

For example, a rotation or reflection in the plane has a familiar standard matrix. But if we describe the plane using a different basis, the entries of the matrix change.

The transformation remains the same. Only its coordinate representation changes.

This distinction prevents a common error: two different matrices may represent the same linear map in different bases, and the same matrix may represent different maps when different bases are being used.

34.11 Change-of-Coordinates Matrices

Let

be a basis of $F^n$. The change-of-coordinates matrix from $B$-coordinates to standard coordinates is

If $x$ is a vector in $F^n$, then

Thus $P_B$ converts coordinates in basis $B$ into standard coordinates.

Since $B$ is a basis, $P_B$ is invertible. Therefore

The inverse matrix converts standard coordinates into $B$-coordinates.

34.12 Matrix Relative to Domain and Codomain Bases

Suppose $T:F^n\to F^m$ has standard matrix $A$. Let $B$ be a basis of $F^n$, and let $C$ be a basis of $F^m$.

Let $P_B$ be the matrix with columns equal to the basis vectors in $B$, and let $P_C$ be the matrix with columns equal to the basis vectors in $C$.

We want a matrix $M$ such that

Since

we have

Now convert the output to $C$-coordinates:

Therefore

So

This formula is the main computational rule for changing matrix representations.

34.13 Operators and Similarity

A linear operator is a linear map from a vector space to itself:

When the same basis $B$ is used for both the domain and codomain, the matrix is written

If $A$ is the standard matrix of a linear operator on $F^n$, and $B$ is a basis with change-of-coordinates matrix $P_B$, then

Matrices related by

are called similar matrices.

Similar matrices represent the same linear operator in different bases. They have many shared properties, including determinant, trace, rank, characteristic polynomial, and eigenvalues.

This is why choosing a good basis matters. A complicated matrix may become simpler after a change of basis.

34.14 Diagonal Representation

A particularly useful case occurs when a linear operator has a basis of eigenvectors.

Let

have basis

such that

for each $j$. Then the matrix of $T$ in basis $B$ is diagonal:

The reason is simple. The $j$-th column of $[T]_B$ is the coordinate vector of $T(v_j)$ in basis $B$. Since

that coordinate vector has $\lambda_j$ in position $j$ and zeros elsewhere.

Diagonalization is the process of finding such a basis when it exists.

34.15 Composition of Linear Maps

Matrix representation converts composition of linear maps into multiplication of matrices.

Let

and

be linear maps. Choose bases $A$ for $U$, $B$ for $V$, and $C$ for $W$.

Then

represents $T$, and

represents $S$.

The composition

has matrix

The order of multiplication follows the order of application: first $T$, then $S$. With column vectors, the first transformation appears on the right. Matrix multiplication represents composition of linear transformations.

34.16 Inverses

If $T:V\to W$ is an isomorphism, then $T^{-1}:W\to V$ exists and is linear.

Choose bases $B$ for $V$ and $C$ for $W$. If

then the matrix of the inverse map is

Indeed,

In coordinates,

Therefore

So

A linear map is invertible exactly when its representing matrix is invertible.

34.17 Kernel and Image in Matrix Form

Let

Then the kernel of $T$ corresponds to the null space of $M$ in coordinates:

Thus

Similarly, the image of $T$ corresponds to the column space of $M$ in codomain coordinates:

This means that row reduction and matrix methods can be used to compute bases for kernels and images of abstract linear maps, once bases have been chosen.

34.18 Example: A Polynomial Map

Let $P_2$ be the vector space of polynomials of degree at most $2$, and let $P_1$ be the vector space of polynomials of degree at most $1$.

Define

by differentiation:

Choose the ordered basis

for $P_2$, and

for $P_1$.

Compute the images of the basis vectors:

Now write each image in the basis $C$:

Therefore

If

then

Multiplying,

This is the coordinate vector of

in the basis $C=(1,x)$.

34.19 Example: Integration as a Linear Map

Let $P_1$ be the vector space of polynomials of degree at most $1$, and let $P_2$ be the vector space of polynomials of degree at most $2$.

Define

by

Use bases

for $P_1$, and

for $P_2$.

Compute:

and

Therefore

So

If

then

Multiplying gives

which represents

34.20 Reading a Matrix as a Linear Map

A matrix can be read column by column.

If

then $A$ represents a map

The columns are

For

we have

Thus a matrix does not merely multiply numbers. It forms a linear combination of its columns using the input coordinates as coefficients.

This column view explains why the image of $A$ is the span of its columns.

34.21 Row View

The row view is also useful.

If $A$ is an $m\times n$ matrix, then each row defines a linear functional on $F^n$. The output vector $Ax$ contains the dot products of the rows of $A$ with $x$.

For

where each $r_i$ is a row vector,

The row view is useful for systems of equations. The equation

means that each row imposes one linear equation on $x$.

The column view emphasizes the image. The row view emphasizes constraints.

34.22 Standard Geometric Matrices

Many geometric transformations in the plane are represented by simple matrices.

A scaling transformation is

A reflection across the $x$-axis is

A projection onto the $x$-axis is

A shear parallel to the $x$-axis is

A rotation through angle $\theta$ is

These matrices act on column vectors in $\mathbb{R}^2$. They are linear because they keep the origin fixed and preserve linear combinations.

34.23 Common Mistakes

The first common mistake is reversing the matrix size. A map

has an $m\times n$ matrix, not an $n\times m$ matrix.

The second common mistake is putting images of basis vectors in rows instead of columns. With column vectors, the image of the $j$-th basis vector goes in the $j$-th column.

The third common mistake is forgetting the codomain basis. To form $[T]_{C\leftarrow B}$, compute $T(v_j)$, then express it in the basis $C$.

The fourth common mistake is treating a matrix as basis-independent. A matrix represents a linear map only after the relevant bases are known.

The fifth common mistake is using changed columns from a row-reduced matrix as a basis for the original image. Row reduction helps identify pivot columns, but the corresponding basis vectors for the image must be taken from the original matrix.

34.24 Summary

A matrix representation gives coordinates for a linear map.

For a linear map

with ordered basis $B=(v_1,\ldots,v_n)$ of $V$ and ordered basis $C=(w_1,\ldots,w_m)$ of $W$, the matrix of $T$ is

It satisfies

In standard coordinate spaces, the matrix of $T:\mathbb{R}^n\to\mathbb{R}^m$ has columns $T(e_1),\ldots,T(e_n)$, and

Composition of linear maps becomes matrix multiplication. Inverses of linear maps become inverse matrices. A change of basis changes the matrix representation by multiplication with change-of-coordinates matrices.

The linear map is the coordinate-free object. The matrix is its coordinate description.