LA-4.1

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Matrix Representation

Every linear map between finite-dimensional spaces can be represented by a matrix—once we choose bases. This fundamental correspondence bridges abstract linear algebra with concrete numerical computation.

3-4 hours Core Level 10 Objectives

Learning Objectives

Understand the correspondence between linear maps and matrices
Compute the matrix representation of a linear map given bases
Apply the column-by-column construction method
Verify that matrix-vector multiplication computes coordinates of the image
Define and compute change of basis matrices
Apply the similarity transformation formula
Recognize that similar matrices represent the same linear map
Calculate the matrix of composed linear maps
Understand the isomorphism between ℒ(V, W) and F^{m×n}
Work with matrices of standard transformations (rotation, projection, etc.)

Prerequisites

Linear map definition (LA-3.1)
Basis and dimension (LA-2.4)
Coordinate vectors
Matrix multiplication basics
Linear independence

Historical Context

The connection between linear maps and matrices was gradually understood through the 19th century. Arthur Cayley (1821–1895) developed matrix algebra in 1858, treating matrices as algebraic objects in their own right. He recognized that matrix multiplication corresponds to composition of linear transformations.

The modern perspective—that matrices are representations of abstract linear maps—emerged in the early 20th century with the axiomatization of vector spaces. This view emphasizes that the matrix depends on the choice of basis, while the underlying linear map does not.

Understanding this representation is crucial: it allows us to compute with linear maps using matrix arithmetic, while recognizing that properties like rank and eigenvalues are intrinsic to the map itself, independent of any particular matrix representation.

1. From Linear Maps to Matrices

We begin with a fundamental question: how can we represent abstract linear maps with concrete numbers? The key insight is that a linear map is completely determined by its action on a basis. Once we choose bases for the domain and codomain, we can record the images of basis vectors as columns of a matrix.

Remark 4.1: Motivation: Maps on Fⁿ

Consider first the special case of linear maps $\sigma: F^n \to F^m$ . Any such map satisfies:

\sigma(x) = \sigma(x_1 e_1 + \cdots + x_n e_n) = x_1 \sigma(e_1) + \cdots + x_n \sigma(e_n)

If we let $\sigma(e_j) = (a_{1j}, a_{2j}, \ldots, a_{mj})^T$ , then $\sigma(x) = Ax$ where $A$ is the matrix with columns $\sigma(e_1), \ldots, \sigma(e_n)$ .

Theorem 4.1: Unique Matrix for Maps on Fⁿ

Any linear map $\sigma: F^n \to F^m$ can be written as $\sigma(x) = Ax$ for a unique $m \times n$ matrix $A$ .

Proof:

Existence: Define $A$ by setting column $j$ equal to $\sigma(e_j)$ . Then by linearity:

\sigma(x) = \sum_{j=1}^{n} x_j \sigma(e_j) = Ax

Uniqueness: If $Ax = Bx$ for all $x$ , taking $x = e_j$ shows that columns $j$ of $A$ and $B$ are equal.

∎

For abstract vector spaces $V$ and $W$ , we use coordinate maps to reduce to the case above. Given bases, the coordinate map $\phi_B: V \to F^n$ is an isomorphism, and we can represent any linear map via the commutative diagram:

Definition 4.1: Matrix of a Linear Map

Let $\sigma \in L(V_1, V_2)$ where $\dim V_1 = n$ and $\dim V_2 = m$ . Let $B_1 = \{\varepsilon_1, \ldots, \varepsilon_n\}$ be a basis of $V_1$ and $B_2 = \{\alpha_1, \ldots, \alpha_m\}$ be a basis of $V_2$ .

Since each $\sigma(\varepsilon_j) \in V_2$ , we can write:

\sigma(\varepsilon_j) = a_{1j}\alpha_1 + a_{2j}\alpha_2 + \cdots + a_{mj}\alpha_m

The matrix representation of $\sigma$ with respect to $B_1$ and $B_2$ is:

M(\sigma) = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}

Column $j$ contains the coordinates of $\sigma(\varepsilon_j)$ in basis $B_2$ .

Remark 4.2: Compact Notation

We often write the definition compactly as:

(\sigma(\varepsilon_1), \sigma(\varepsilon_2), \ldots, \sigma(\varepsilon_n)) = (\alpha_1, \alpha_2, \ldots, \alpha_m) M(\sigma)

This notation emphasizes that we multiply the row of basis vectors by the matrix to get the row of images.

Remark 4.3: Matrix Dimensions

The matrix $M(\sigma)$ is $m \times n$ where:

$m$ = dimension of the codomain (arrival space)
$n$ = dimension of the domain (departure space)

Note the reversal: rows correspond to the codomain, columns to the domain.

2. Coordinate Transformation Formula

The matrix representation has a crucial property: it transforms coordinates. If we know the coordinates of a vector in the domain basis, matrix multiplication gives us the coordinates of its image in the codomain basis.

Theorem 4.2: Coordinate Transformation

Let $\sigma \in L(V_1, V_2)$ have matrix $A = M(\sigma)$ with respect to bases $B_1$ and $B_2$ . If $\alpha \in V_1$ has coordinates $X$ (in $B_1$ ) and $\sigma(\alpha)$ has coordinates $Y$ (in $B_2$ ), then:

Y = AX

Proof:

Let $\alpha = x_1 \varepsilon_1 + \cdots + x_n \varepsilon_n$ . By linearity:

\sigma(\alpha) = x_1 \sigma(\varepsilon_1) + \cdots + x_n \sigma(\varepsilon_n)

Substituting the expressions for $\sigma(\varepsilon_j)$ in terms of $B_2$ :

\sigma(\alpha) = \sum_{j=1}^n x_j \sum_{i=1}^m a_{ij} \alpha_i = \sum_{i=1}^m \left(\sum_{j=1}^n a_{ij} x_j\right) \alpha_i

The coefficient of $\alpha_i$ is $\sum_j a_{ij} x_j$ , which is the $i$ -th entry of $AX$ .

∎

Corollary 4.1: Matrix-Map Correspondence

For fixed bases, there is a one-to-one correspondence between linear maps $L(V_1, V_2)$ and matrices $F^{m \times n}$ . Each linear map has a unique matrix, and each matrix determines a unique linear map.

3. The Isomorphism Between Linear Maps and Matrices

The matrix representation is more than just a correspondence—it is an isomorphism of vector spaces. This means we can transfer questions about linear maps to questions about matrices.

Theorem 4.3: Linear Maps and Matrices are Isomorphic

Let $V_1$ and $V_2$ be vector spaces of dimensions $n$ and $m$ over $F$ . Then:

L(V_1, V_2) \cong F^{m \times n}

In particular, $\dim L(V_1, V_2) = mn$ .

Proof:

The matrix representation $\phi: L(V_1, V_2) \to F^{m \times n}$ defined by $\phi(\sigma) = M(\sigma)$ is:

Linear: $M(\sigma + \tau) = M(\sigma) + M(\tau)$ and $M(\lambda\sigma) = \lambda M(\sigma)$
Injective: If $M(\sigma) = 0$ , then $\sigma(\varepsilon_j) = 0$ for all basis vectors, so $\sigma = 0$
Surjective: Given any matrix $A$ , define $\sigma(\varepsilon_j) = \sum_i a_{ij}\alpha_i$ to get a map with $M(\sigma) = A$

∎

Remark 4.4: Basis of the Matrix Space

The space $F^{m \times n}$ has a natural basis: the matrices $E_{ij}$ with a 1 in position $(i,j)$ and 0 elsewhere. There are $mn$ such matrices.

Example 4.1: Dimension Calculation

$\dim L(\mathbb{R}^3, \mathbb{R}^2) = 2 \times 3 = 6$
$\dim L(P_4, \mathbb{R}^3) = 3 \times 5 = 15$ (since $\dim P_4 = 5$ )
$\dim L(\mathbb{R}^n, \mathbb{R}^n) = n^2$

4. Computing Matrix Representations

Example 4.2: Linear Map on ℝ³

Problem: Let $\sigma \in L(\mathbb{R}^3, \mathbb{R}^3)$ with $\sigma(x_1, x_2, x_3)^T = (x_1 + x_2, x_1 - x_3, x_2)^T$ .

Find the matrix in the standard basis.

Solution: Compute the images of basis vectors:

\sigma(e_1) = \sigma(1,0,0)^T = (1, 1, 0)^T

\sigma(e_2) = \sigma(0,1,0)^T = (1, 0, 1)^T

\sigma(e_3) = \sigma(0,0,1)^T = (0, -1, 0)^T

The matrix is formed by placing these as columns:

M(\sigma) = \begin{pmatrix} 1 & 1 & 0 \\ 1 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}

Example 4.3: Rotation in ℝ²

Problem: Find the matrix of rotation by angle $\theta$ counterclockwise.

Solution: The rotation $R_\theta$ sends:

R_\theta(e_1) = (\cos\theta, \sin\theta)^T

R_\theta(e_2) = (-\sin\theta, \cos\theta)^T

Therefore:

M(R_\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Example 4.4: Projection onto a Line

Problem: Find the matrix of orthogonal projection onto the line $y = x$ in $\mathbb{R}^2$ .

Solution: The projection formula onto a unit vector $u = (1/\sqrt{2}, 1/\sqrt{2})^T$ is $P(v) = (u \cdot v)u$ .

P(e_1) = \frac{1}{2}(1, 1)^T, \quad P(e_2) = \frac{1}{2}(1, 1)^T

The projection matrix is:

M(P) = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}

Example 4.5: Differentiation Operator

Problem: Find the matrix of $D: P_3 \to P_2$ where $D(p) = p'$ , using standard bases.

Solution: Basis of $P_3$ : $\{1, x, x^2, x^3\}$ . Basis of $P_2$ : $\{1, x, x^2\}$ .

D(1) = 0, \quad D(x) = 1, \quad D(x^2) = 2x, \quad D(x^3) = 3x^2

In coordinates:

M(D) = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \end{pmatrix}

5. Change of Basis

What happens to the matrix when we change bases? This is fundamental for finding nice representations.

Definition 4.2: Change of Basis Matrix

Let $B$ and $B'$ be bases of $V$ . The change of basis matrix from $B'$ to $B$ is the matrix $P$ such that:

[v]_B = P[v]_{B'}

Column $j$ of $P$ contains the coordinates of the $j$ -th vector of $B'$ in basis $B$ .

Theorem 4.4: Change of Basis for Linear Maps

Let $\sigma \in L(V_1, V_2)$ have matrix $A$ with respect to bases $B_1, B_2$ . If we change to new bases $B_1', B_2'$ , the new matrix is:

A' = Q^{-1}AP

where $P$ is the change from $B_1'$ to $B_1$ , and $Q$ is the change from $B_2'$ to $B_2$ .

Proof:

For any $v \in V_1$ , we have $[\sigma(v)]_{B_2} = A[v]_{B_1}$ . Using $[v]_{B_1} = P[v]_{B_1'}$ and $[\sigma(v)]_{B_2} = Q[\sigma(v)]_{B_2'}$ :

Q[\sigma(v)]_{B_2'} = AP[v]_{B_1'}

Therefore $[\sigma(v)]_{B_2'} = Q^{-1}AP[v]_{B_1'}$ , so $A' = Q^{-1}AP$ .

∎

Corollary 4.2: Similar Matrices (for Operators)

For a linear operator $\sigma: V \to V$ , if we use the same basis for domain and codomain, changing to a new basis $B'$ gives:

A' = P^{-1}AP

Two matrices related by $B = P^{-1}AP$ are called similar.

Remark 4.5: Properties Preserved by Similarity

Similar matrices share many properties (they represent the same operator!):

Determinant
Trace
Eigenvalues
Characteristic polynomial
Rank and nullity

6. Worked Examples

Example 1: Computing Image and Kernel

Problem: For $\sigma(x_1, x_2, x_3)^T = (x_1 + x_2, x_1 - x_3, x_2)^T$ , find $\text{im } \sigma$ and $\ker \sigma$ .

Solution:

\text{im } \sigma = \text{span}\{\sigma(e_1), \sigma(e_2), \sigma(e_3)\} = \text{span}\{(1,1,0)^T, (1,0,1)^T, (0,-1,0)^T\}

These three vectors span $\mathbb{R}^3$ , so $\text{im } \sigma = \mathbb{R}^3$ and $\sigma$ is surjective.

For the kernel, solve $\sigma(\alpha) = 0$ :

x_1 + x_2 = 0, \quad x_1 - x_3 = 0, \quad x_2 = 0

This gives $x_1 = x_2 = x_3 = 0$ , so $\ker \sigma = \{0\}$ .

Example 2: Change of Basis

Problem: The reflection about the line $y = x$ has matrix $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ in the standard basis. Find its matrix in the basis $B' = \{(1,1)^T, (1,-1)^T\}$ .

Solution: The change of basis matrix from $B'$ to standard is:

P = \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}, \quad P^{-1} = \frac{1}{-2}\begin{pmatrix} -1 & -1 \\ -1 & 1 \end{pmatrix} = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & -1/2 \end{pmatrix}

The new matrix is:

A' = P^{-1}AP = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

The matrix is diagonal! The basis vectors are eigenvectors.

Example 3: Matrix of Composition

Problem: Let $T(x,y) = (x+y, x)$ and $S(x,y) = (2x, y-x)$ . Find $[S \circ T]$ .

Solution: First, find the individual matrices:

[T] = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}, \quad [S] = \begin{pmatrix} 2 & 0 \\ -1 & 1 \end{pmatrix}

Then multiply (in the correct order!):

[S \circ T] = [S][T] = \begin{pmatrix} 2 & 0 \\ -1 & 1 \end{pmatrix}\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 2 \\ 0 & -1 \end{pmatrix}

Example 4: Non-Standard Basis

Problem: Find the matrix of $D: P_2 \to P_1$ (differentiation) using basis $\{1, x-1, (x-1)^2\}$ for $P_2$ and $\{1, x-1\}$ for $P_1$ .

Solution: Compute derivatives:

D(1) = 0 = 0 \cdot 1 + 0 \cdot (x-1)

D(x-1) = 1 = 1 \cdot 1 + 0 \cdot (x-1)

D((x-1)^2) = 2(x-1) = 0 \cdot 1 + 2 \cdot (x-1)

The matrix is:

M(D) = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix}

7. Common Mistakes

Mistake 1: Wrong Matrix Dimensions

A map $T: V \to W$ with $\dim V = n$ and $\dim W = m$ has an $m \times n$ matrix, NOT $n \times m$ . Rows = codomain, columns = domain.

Mistake 2: Row vs Column Confusion

Columns of the matrix are the images of basis vectors, not rows. Writing $\sigma(e_j)$ as a row instead of a column produces the transpose of the correct matrix.

Mistake 3: Composition Order

For $S \circ T$ , the matrix is $[S][T]$ , not $[T][S]$ . The rightmost matrix acts first on the vector.

Mistake 4: Change of Basis Formula

For operators, the formula is $A' = P^{-1}AP$ , not $PAP^{-1}$ or $P^{-1}A$ . The order of $P$ and $P^{-1}$ matters!

Mistake 5: Forgetting the Basis

A matrix alone is meaningless—you must specify the bases. The same linear map has different matrices in different bases. Always track which bases are being used.

8. Key Takeaways

Column Construction

Column $j$ of $[T]_B^C$ is the coordinate vector $[T(b_j)]_C$ . Apply $T$ to each domain basis vector and express the result in the codomain basis.

Coordinate Transformation

$[T(v)]_C = [T]_B^C [v]_B$ . The matrix transforms input coordinates to output coordinates via matrix-vector multiplication.

Isomorphism

$L(V_1, V_2) \cong F^{m \times n}$ . Linear maps and matrices are in one-to-one correspondence (for fixed bases).

Change of Basis

$A' = Q^{-1}AP$ (general) or $A' = P^{-1}AP$ (operators). Similar matrices represent the same linear map in different bases.

9. Additional Practice Problems

Problem 1

Find the matrix of $T: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $T(x, y) = (x + y, x - y, 2x)$ in the standard bases.

Problem 2

Let $A$ be the matrix of $T$ in standard basis. Find the matrix of $T$ in the basis $B = \{(1, 1), (1, -1)\}$ .

Problem 3

Find the matrix of the linear operator $T: M_{2 \times 2} \to M_{2 \times 2}$ defined by $T(A) = A^T$ using the standard basis $\{E_{11}, E_{12}, E_{21}, E_{22}\}$ .

Problem 4

Show that the matrices $\begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ and $\begin{pmatrix} 3 & 0 \\ 2 & 1 \end{pmatrix}$ are similar by finding a matrix $P$ .

Problem 5

Find the matrix of the integration operator $I: P_2 \to P_3$ defined by $I(p)(x) = \int_0^x p(t) dt$ .

Problem 6

Prove that if $A \in M_n(F)$ is similar to a diagonal matrix, then $A^k$ is also similar to a diagonal matrix for all $k \geq 1$ .

10. Connections to Other Topics

Matrix representation is the bridge between abstract linear algebra and computation. It connects to virtually every other topic in linear algebra.

Kernel and Image

The kernel of $T$ corresponds to the null space of its matrix $A$ : solutions to $Ax = 0$ . The image of $T$ is the column space of $A$ . These correspondences allow us to compute kernels and images using row reduction.

Rank-Nullity Theorem

For the matrix $A$ of $T: V \to W$ :

\text{rank}(A) + \text{nullity}(A) = n = \dim V

The rank equals the number of pivot columns, and nullity equals the number of free variables.

Determinants

For a linear operator $T: V \to V$ , the determinant $\det(A)$ is independent of the choice of basis (similar matrices have equal determinants). It measures the "volume scaling factor" of the transformation.

Eigenvalues and Diagonalization

If we can find a basis of eigenvectors, the matrix in that basis is diagonal. Finding the "right" basis to simplify the matrix is a central theme: eigenvalue decomposition, Jordan form, and SVD are all about choosing bases to reveal structure.

Inner Product Spaces

In an orthonormal basis, inner products become dot products: $\langle u, v \rangle = [u]^T [v]$ . The adjoint $T^*$ has matrix $A^* = \bar{A}^T$ . Self-adjoint operators have real eigenvalues and orthogonal eigenvectors.

11. Theoretical Insights

Theorem 4.5: Matrix Representation Preserves Operations

The matrix representation respects linear algebra operations:

$M(\sigma + \tau) = M(\sigma) + M(\tau)$
$M(\lambda \sigma) = \lambda M(\sigma)$
$M(\tau \circ \sigma) = M(\tau) M(\sigma)$ (when composition is defined)
$M(\text{id}_V) = I_n$
$M(\sigma^{-1}) = M(\sigma)^{-1}$ (when $\sigma$ is invertible)

Proof:

(1) and (2): Follow from the linearity of the representation map.

(3): For any $v \in V$ :

[\tau(\sigma(v))]_C = M(\tau)[\sigma(v)]_B = M(\tau)M(\sigma)[v]_A

This equals $[(\tau \circ \sigma)(v)]_C = M(\tau \circ \sigma)[v]_A$ .

(4): $\text{id}(v_j) = v_j$ has coordinates $e_j$ , so column $j$ is $e_j$ .

(5): From $\sigma \circ \sigma^{-1} = \text{id}$ , we get $M(\sigma)M(\sigma^{-1}) = I$ .

∎

Theorem 4.6: Invariance Under Change of Basis

The following properties of a linear operator $T: V \to V$ are independent of the choice of basis:

Rank and nullity
Determinant
Trace
Characteristic polynomial
Minimal polynomial
Eigenvalues (with multiplicities)

Remark 4.6: Intrinsic vs Extrinsic Properties

Properties that don't depend on the basis choice are called intrinsic—they belong to the linear map itself. Properties that depend on the basis (like individual matrix entries) are extrinsic. The goal of much of linear algebra is to find intrinsic properties and canonical forms.

Corollary 4.3: Similar Matrices Have Same Trace and Determinant

If $B = P^{-1}AP$ , then:

\text{tr}(B) = \text{tr}(A), \quad \det(B) = \det(A)

Proof:

For trace: $\text{tr}(P^{-1}AP) = \text{tr}(APP^{-1}) = \text{tr}(A)$ using $\text{tr}(XY) = \text{tr}(YX)$ .

For determinant: $\det(P^{-1}AP) = \det(P^{-1})\det(A)\det(P) = \det(A)$ .

∎

12. Geometric Interpretation

Matrix representation gives concrete geometric meaning to linear transformations.

Columns as Transformed Basis Vectors

The columns of a matrix show where the standard basis vectors go. For a $2 \times 2$ matrix:

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

Column 1 is $(a, c)^T = A e_1$ : where $e_1 = (1, 0)^T$ gets mapped.
Column 2 is $(b, d)^T = A e_2$ : where $e_2 = (0, 1)^T$ gets mapped.

To visualize the transformation, draw the unit square and see where it maps—the columns are the images of the two edges from the origin.

Standard Transformations

Rotation by θ:

R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Reflection about x-axis:

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

Scaling by factors a and b:

\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}

Shear (horizontal):

\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}

Projection onto x-axis:

\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}

Determinant as Area/Volume Factor

For a $2 \times 2$ matrix, $|\det(A)|$ is the factor by which areas scale. For $3 \times 3$ , it's the volume scaling factor. A negative determinant indicates orientation reversal (like a reflection).

If $\det(A) = 0$ , the transformation collapses dimension—the image is a lower-dimensional subspace (like projecting 3D to a plane).

13. Challenge Problems

Challenge 1: Commuting Matrices

Let $A, B \in M_n(F)$ . Prove that if $AB = BA$ , then for any polynomial $p$ :

p(A)B = Bp(A)

Hint: First prove for $p(x) = x^k$ by induction, then extend by linearity.

Challenge 2: Trace and Commutators

Prove that for any $A, B \in M_n(F)$ :

\text{tr}(AB - BA) = 0

Conclude that no matrices $A$ , $B$ satisfy $AB - BA = I$ .

Challenge 3: Rank and Similarity

Let $A \in M_{m \times n}(F)$ and $B \in M_{n \times m}(F)$ . Prove:

\text{rank}(AB) \leq \min(\text{rank}(A), \text{rank}(B))

Hint: Think about the column space of $AB$ relative to that of $A$ .

14. Study Tips

Think Column-by-Column

Always construct matrices one column at a time. Each column answers: "Where does this basis vector go, expressed in the target basis?"

Check Dimensions First

Before computing, verify that dimensions match. A map $V \to W$ needs an $(\dim W) \times (\dim V)$ matrix.

Verify with Test Vectors

After computing a matrix, verify by testing: compute $T(v)$ directly and via $A[v]_B$ . They should match.

Track Your Bases

Always write down which bases you're using. Notations like $[T]_B^C$ help prevent confusion between different matrix representations.

15. Quick Reference Summary

Concept	Formula/Description
Matrix Representation	Column $j$ = $[T(v_j)]_C$
Matrix Dimensions	$m \times n$ where $m = \dim W$ , $n = \dim V$
Coordinate Transform	$[T(v)]_C = [T]_B^C [v]_B$
Isomorphism	$L(V, W) \cong F^{m \times n}$ , $\dim = mn$
Change of Basis (General)	$A' = Q^{-1}AP$
Change of Basis (Operator)	$A' = P^{-1}AP$
Matrix of Composition	$[S \circ T] = [S][T]$
Similar Matrices	Same det, trace, eigenvalues, char. poly, rank

Historical Notes

The Development of Matrix Notation: The word "matrix" was coined by James Joseph Sylvester in 1850, derived from the Latin word for "womb" (as a matrix is a rectangular array from which determinants can be "born"). Arthur Cayley then developed matrix algebra in his 1858 paper "A Memoir on the Theory of Matrices."

From Equations to Transformations: Initially, matrices were viewed as abbreviations for systems of linear equations. The shift to viewing them as representations of linear transformations came later, influenced by the work of mathematicians like Hermann Grassmann and Giuseppe Peano on abstract vector spaces.

The Cambridge School: Cayley and Sylvester, working in Cambridge, laid the foundations of matrix theory. They discovered that matrices form a ring under addition and multiplication, and that matrix multiplication is generally non-commutative—a surprising departure from ordinary number arithmetic.

Modern Perspective: The 20th century saw the abstraction of linear algebra, with emphasis on vector spaces defined axiomatically. In this view, matrices are representations that depend on basis choice—a powerful perspective that unifies seemingly different matrices as representations of the same abstract linear map.

More Worked Examples

Example: Left Shift Operator

Problem: Find the matrix of the left shift operator $L: F^4 \to F^4$ defined by $L(x_1, x_2, x_3, x_4) = (x_2, x_3, x_4, 0)$ .

Solution: Compute images of basis vectors:

L(e_1) = (0, 0, 0, 0), \quad L(e_2) = (1, 0, 0, 0)

L(e_3) = (0, 1, 0, 0), \quad L(e_4) = (0, 0, 1, 0)

The matrix is:

M(L) = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}

This is a nilpotent matrix: $L^4 = 0$ .

Example: Trace as Linear Functional

Problem: Find the matrix of $\text{tr}: M_{2 \times 2}(\mathbb{R}) \to \mathbb{R}$ .

Solution: Use basis $\{E_{11}, E_{12}, E_{21}, E_{22}\}$ for $M_{2 \times 2}$ and $\{1\}$ for $\mathbb{R}$ .

\text{tr}(E_{11}) = 1, \quad \text{tr}(E_{12}) = 0, \quad \text{tr}(E_{21}) = 0, \quad \text{tr}(E_{22}) = 1

The matrix (a $1 \times 4$ row vector) is:

M(\text{tr}) = \begin{pmatrix} 1 & 0 & 0 & 1 \end{pmatrix}

Example: Coordinate Change Verification

Problem: Verify that $T(x, y) = (2x + y, x - y)$ gives the same result via direct computation and matrix multiplication.

Solution: Matrix in standard basis: $A = \begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix}$

For $v = (3, 2)^T$ :

T(3, 2) = (2(3) + 2, 3 - 2) = (8, 1)

A \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} = \begin{pmatrix} 8 \\ 1 \end{pmatrix} \checkmark

What's Next?

Now that you understand how to represent linear maps as matrices, the next topics build on this foundation:

Matrix Operations: Addition, scalar multiplication, and the crucial matrix product
Matrix Inverse: When does a matrix have an inverse, and how to compute it
Elementary Matrices: Building blocks for matrix factorizations
Matrix Rank: The fundamental invariant connecting matrices to linear maps

These concepts transform abstract linear algebra into computational techniques that power everything from computer graphics to machine learning.

Matrix Representation Practice

Questions

Correct

Accuracy

T: \mathbb{R}^2 \to \mathbb{R}^2

is represented by

A

in the standard basis, what represents

T

in a new basis

B

Medium

Not attempted

The matrix of the identity map

I: V \to V

in any basis is:

Easy

Not attempted

\dim V = 3

and

\dim W = 2

, matrices representing

T: V \to W

are:

Easy

Not attempted

The

j

th column of

[T]_B^C

is:

Medium

Not attempted

A

and

B

represent the same linear operator in different bases, they are:

Medium

Not attempted

[S \circ T]_B^D = ?

where

T: V \to W

and

S: W \to U

Hard

Not attempted

The change of basis matrix from

B

C

has columns that are:

Medium

Not attempted

P

is a change of basis matrix, is

P

always invertible?

Easy

Not attempted

\sigma \in L(\mathbb{R}^3, \mathbb{R}^3)

with

\sigma(x_1,x_2,x_3) = (x_1+x_2, x_1-x_3, x_2)

, the matrix in standard basis is:

Medium

Not attempted

The rotation matrix

R_\theta

\mathbb{R}^2

by angle

\theta

is:

Easy

Not attempted

\dim V_1 = n

and

\dim V_2 = m

, then

L(V_1, V_2) \cong

Medium

Not attempted

For

\sigma: V_1 \to V_2

with matrix

A

, if

\alpha

has coordinates

X

and

\sigma(\alpha)

has coordinates

Y

, then:

Easy

Not attempted

Frequently Asked Questions

Why does the matrix of a linear map depend on the choice of basis?

The matrix encodes how the linear map transforms coordinates. Different bases mean different coordinate systems, so the same abstract map gets different numerical representations. The key insight is that while the matrix changes, the intrinsic properties of the map (rank, nullity, eigenvalues) remain unchanged.

What is the relationship between similar matrices?

Similar matrices represent the same linear operator in different bases. They share many properties: determinant, trace, eigenvalues, characteristic polynomial, and rank. If $B = P^{-1}AP$, then $A$ and $B$ are similar, connected by the change of basis matrix $P$.

How do I compute a change of basis matrix?

Write each vector of the new basis as a linear combination of the old basis vectors. The coefficients form the columns of the change of basis matrix. For example, if $b_j = \sum_i p_{ij} c_i$, then column $j$ of $P$ contains the coefficients $p_{1j}, p_{2j}, \ldots$.

What is the 'best' basis to use?

It depends on the problem. For eigenvalue problems, an eigenbasis diagonalizes the matrix. For inner product calculations, an orthonormal basis simplifies computations. For computational efficiency, sparse or structured bases may be preferred.

Is there a coordinate-free way to do linear algebra?

Yes! The abstract approach works without coordinates—defining linear maps, kernels, images, etc. without choosing bases. However, for numerical computation, we eventually need bases and matrices. The power of the theory is that results (like rank) don't depend on the arbitrary basis choice.

Why is the matrix $m \times n$ when mapping from $n$-dim to $m$-dim?

The matrix must multiply an $n$-dimensional coordinate vector (column with $n$ entries) and produce an $m$-dimensional result. For this multiplication to work, the matrix must have $m$ rows and $n$ columns: $(m \times n) \cdot (n \times 1) = (m \times 1)$.

How does matrix multiplication relate to function composition?

Matrix multiplication corresponds exactly to composition of linear maps. If $T$ has matrix $A$ and $S$ has matrix $B$, then $S \circ T$ has matrix $BA$ (note the order!). This is why matrix multiplication is associative but not commutative.

What's the difference between $[T]_B$ and $[T]_B^C$?

The notation $[T]_B^C$ denotes the matrix of $T: V \to W$ using basis $B$ for the domain $V$ and basis $C$ for the codomain $W$. When $T: V \to V$ is an operator and we use the same basis for both, we often write $[T]_B$ as shorthand for $[T]_B^B$.

Can two different linear maps have the same matrix?

Yes, but only if we use different bases! For fixed bases, the correspondence between linear maps and matrices is one-to-one (an isomorphism). Two maps with the same matrix in the same bases are identical.

How do I verify my matrix representation is correct?

Check that for each basis vector $v_j$, the $j$th column equals $[T(v_j)]_C$ (the coordinates of the image in the codomain basis). You can also verify by computing $[T(v)]_C = A[v]_B$ for a few test vectors.