Linear Transformations

⚡ Key Takeaways

A linear transformation is a function $T: V \to W$ between vector spaces that preserves vector addition and scalar multiplication. Key result: every linear transformation between finite-dimensional spaces (once bases are chosen) corresponds to a unique matrix, and matrix multiplication encodes composition. Applies to: geometry, differential equations, quantum mechanics, computer graphics, data science. Foundational to all of linear algebra; formalized in the 19th–20th centuries by Cayley, Grassmann, Peano, and Noether.

💡 The Core Insight

A linear transformation is completely determined by its action on a basis. If you know where each basis vector goes, you know where every vector goes, because every vector is a unique linear combination of basis vectors and $T$ commutes with linear combinations. This single fact collapses an infinite amount of information into a finite table of numbers — the matrix of the transformation — and turns the study of functions into the study of arrays.

DefinitionLinear Transformation

Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$ . A function $T: V \to W$ is a linear transformation (or linear map) if for all $\mathbf{u}, \mathbf{v} \in V$ and all scalars $c \in \mathbb{F}$ :

1

\quad T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})

Derivation hint: this is the additivity axiom; it states $T$ is a homomorphism of the underlying abelian groups.

2

\quad T(c\,\mathbf{v}) = c\,T(\mathbf{v})

Derivation hint: this is the homogeneity axiom; combined with [1] it makes $T$ respect all linear combinations. Here $\mathbf{u},\mathbf{v}$ are vectors in the domain $V$ , $c$ is a scalar from $\mathbb{F}$ , and $T(\mathbf{u}), T(\mathbf{v})$ live in the codomain $W$ . Both axioms combine into the single superposition principle:

3

\quad T(c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n) = c_1 T(\mathbf{v}_1) + c_2 T(\mathbf{v}_2) + \cdots + c_n T(\mathbf{v}_n)

Derivation hint: apply [1] repeatedly to split the sum, then [2] to pull each scalar out; proven by induction on $n$ .

TheoremMatrix Representation Theorem

Let $V$ be an $n$ -dimensional vector space with ordered basis $\mathcal{B} = (\mathbf{b}_1, \ldots, \mathbf{b}_n)$ and $W$ an $m$ -dimensional space with ordered basis $\mathcal{C} = (\mathbf{c}_1, \ldots, \mathbf{c}_m)$ . Then every linear transformation $T: V \to W$ is represented by a unique $m \times n$ matrix $A = [T]_{\mathcal{C}}^{\mathcal{B}}$ such that

4

\quad [T(\mathbf{v})]_{\mathcal{C}} = A\,[\mathbf{v}]_{\mathcal{B}}

Derivation hint: the $j$ -th column of $A$ is $[T(\mathbf{b}_j)]_{\mathcal{C}}$ , the coordinate vector of the image of the $j$ -th basis vector; uniqueness follows since coordinate maps are bijections. Moreover, composition of linear maps corresponds to matrix multiplication: if $S: W \to U$ has matrix $B$ , then $S \circ T$ has matrix $BA$ .

DefinitionKernel and Image

For a linear transformation $T: V \to W$ , the kernel (null space) and image (range) are:

5

\quad \ker(T) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}, \qquad \operatorname{im}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\}

Derivation hint: both are subspaces — closure under addition and scalar multiplication follows directly from linearity [1]–[2]. The rank is $\dim(\operatorname{im}(T))$ and the nullity is $\dim(\ker(T))$ . They satisfy the Rank–Nullity Theorem:

6

\quad \dim(V) = \operatorname{rank}(T) + \operatorname{nullity}(T)

Derivation hint: extend a basis of $\ker(T)$ to a basis of $V$ ; the images of the added vectors form a basis of $\operatorname{im}(T)$ .

ProofMatrix Representation — Construction and Uniqueness

We show every linear $T: V \to W$ is captured by a unique matrix.

Step 1 — Coordinates exist. Since $\mathcal{B} = (\mathbf{b}_1,\ldots,\mathbf{b}_n)$ is a basis, any $\mathbf{v} \in V$ has a unique expansion

\mathbf{v} = x_1\mathbf{b}_1 + \cdots + x_n\mathbf{b}_n,

so $[\mathbf{v}]_{\mathcal{B}} = (x_1,\ldots,x_n)^\top$ is well-defined.

Step 2 — Apply linearity. Using superposition [3],

T(\mathbf{v}) = T\!\left(\sum_{j=1}^n x_j \mathbf{b}_j\right) = \sum_{j=1}^n x_j\, T(\mathbf{b}_j).

Thus $T(\mathbf{v})$ is determined entirely by the $n$ vectors $T(\mathbf{b}_1), \ldots, T(\mathbf{b}_n)$ .

Step 3 — Express images in $\mathcal{C}$ . Each $T(\mathbf{b}_j) \in W$ has a coordinate vector $[T(\mathbf{b}_j)]_{\mathcal{C}} = (a_{1j}, \ldots, a_{mj})^\top$ . Define $A$ to have these as its columns: $A_{ij} = a_{ij}$ .

Step 4 — Verify [4]. Taking coordinates of $T(\mathbf{v})$ in $\mathcal{C}$ :

[T(\mathbf{v})]_{\mathcal{C}} = \sum_{j=1}^n x_j [T(\mathbf{b}_j)]_{\mathcal{C}} = \sum_{j=1}^n x_j (\text{column } j \text{ of } A) = A[\mathbf{v}]_{\mathcal{B}}.

Step 5 — Uniqueness. If $A'$ also satisfies [4], then $A[\mathbf{v}]_{\mathcal{B}} = A'[\mathbf{v}]_{\mathcal{B}}$ for all $\mathbf{v}$ . Choosing $\mathbf{v} = \mathbf{b}_j$ gives $[\mathbf{b}_j]_{\mathcal{B}} = \mathbf{e}_j$ , so column $j$ of $A$ equals column $j$ of $A'$ . Hence $A = A'$ . $\blacksquare$

∎

ProofRank–Nullity Theorem

Step 1. Let $\dim V = n$ and let $\{\mathbf{u}_1, \ldots, \mathbf{u}_k\}$ be a basis of $\ker(T)$ , so $\operatorname{nullity}(T) = k$ .

Step 2. Extend to a basis $\{\mathbf{u}_1, \ldots, \mathbf{u}_k, \mathbf{w}_1, \ldots, \mathbf{w}_{n-k}\}$ of $V$ .

Step 3 — Spanning. For any $\mathbf{v} = \sum a_i\mathbf{u}_i + \sum b_j\mathbf{w}_j$ , linearity gives $T(\mathbf{v}) = \sum b_j T(\mathbf{w}_j)$ (kernel terms vanish). So $\{T(\mathbf{w}_1),\ldots,T(\mathbf{w}_{n-k})\}$ spans $\operatorname{im}(T)$ .

Step 4 — Independence. Suppose $\sum c_j T(\mathbf{w}_j) = \mathbf{0}$ . Then $T(\sum c_j\mathbf{w}_j) = \mathbf{0}$ , so $\sum c_j\mathbf{w}_j \in \ker(T)$ , hence $\sum c_j\mathbf{w}_j = \sum d_i\mathbf{u}_i$ . Linear independence of the full basis forces all $c_j = 0$ .

Step 5. Therefore $\{T(\mathbf{w}_j)\}$ is a basis of $\operatorname{im}(T)$ , so $\operatorname{rank}(T) = n - k$ , giving $n = \operatorname{rank}(T) + \operatorname{nullity}(T)$ . $\blacksquare$

∎

mathematician · Arthur Cayley | 1821–1895

Cayley introduced matrices as objects in their own right in his 1858 Memoir on the Theory of Matrices, defining matrix addition, multiplication, and the inverse, and recognizing that matrices compose the way linear substitutions do. He observed that a matrix satisfies its own characteristic equation — the Cayley–Hamilton theorem. Though he worked largely with concrete $2\times2$ and $3\times3$ cases, Cayley's abstraction of "a single quantity" representing a whole system of linear equations was the decisive step toward viewing linear transformations as algebraic entities, paving the way for the modern coordinate-free theory developed later by Peano, Grassmann, and Emmy Noether.

🧠 Intuition

Picture a linear transformation as a way of stretching, rotating, shearing, and projecting space while keeping the origin fixed and keeping grid lines straight, parallel, and evenly spaced. That geometric picture is linearity. Equation [1] says "the image of a sum is the sum of images" — if you add two vectors then transform, you get the same result as transforming each and adding. Equation [2] says scaling commutes with the map — doubling the input doubles the output. The deepest intuition is that a linear map can't introduce curves, can't move the origin, and can't bend straight lines. Lines map to lines (or points). Parallelograms map to parallelograms. This rigidity is exactly what lets a finite matrix capture an infinite function: you only need to track where the "grid corners" — the basis vectors — land, and linearity rigidly fills in everything else.

Figure

Action of a linear map: the unit square (grid corners

\mathbf{e}_1,\mathbf{e}_2

) is sent to a parallelogram spanned by

T(\mathbf{e}_1), T(\mathbf{e}_2)

.

ExampleRotation in the Plane

Given: A counterclockwise rotation $R_\theta: \mathbb{R}^2 \to \mathbb{R}^2$ by $\theta = 90^\circ$ . Find its matrix and apply it to $\mathbf{v} = (3, 1)$ .

Step 1 — Track basis vectors. Rotating $\mathbf{e}_1 = (1,0)$ by $90^\circ$ gives $(0,1)$ . Rotating $\mathbf{e}_2 = (0,1)$ by $90^\circ$ gives $(-1,0)$ .

Step 2 — Assemble columns.

A = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.

Step 3 — Apply to $\mathbf{v}$ .

A\begin{pmatrix}3\\1\end{pmatrix} = \begin{pmatrix} 0\cdot3 + (-1)\cdot1 \\ 1\cdot3 + 0\cdot1 \end{pmatrix} = \begin{pmatrix} -1 \\ 3 \end{pmatrix}.

Step 4 — Check. Length preserved: $\|(3,1)\| = \sqrt{10} = \|(-1,3)\|$ . ✓

\boxed{R_{90^\circ}(3,1) = (-1,\,3)}

ExampleDifferentiation as a Linear Map

Given: Let $V = P_3$ , the space of real polynomials of degree $\le 3$ , with basis $\mathcal{B} = (1, x, x^2, x^3)$ . Find the matrix of the derivative operator $D = \frac{d}{dx}: P_3 \to P_3$ , then apply it to $p(x) = 2 + 5x - x^2 + 4x^3$ .

Step 1 — Differentiate basis vectors. $D(1)=0,\quad D(x)=1,\quad D(x^2)=2x,\quad D(x^3)=3x^2.$

Step 2 — Coordinate vectors of images (in basis $1,x,x^2,x^3$ ): $[0]=(0,0,0,0)$ , $[1]=(1,0,0,0)$ , $[2x]=(0,2,0,0)$ , $[3x^2]=(0,0,3,0)$ .

Step 3 — Assemble matrix.

[D] = \begin{pmatrix} 0&1&0&0\\ 0&0&2&0\\ 0&0&0&3\\ 0&0&0&0 \end{pmatrix}.

Step 4 — Apply to $[p]=(2,5,-1,4)^\top$ .

[D]\begin{pmatrix}2\\5\\-1\\4\end{pmatrix} = \begin{pmatrix}5\\-2\\12\\0\end{pmatrix},

which decodes to $5 - 2x + 12x^2$ .

Step 5 — Verify directly: $\frac{d}{dx}(2 + 5x - x^2 + 4x^3) = 5 - 2x + 12x^2$ . ✓

\boxed{D\,p = 5 - 2x + 12x^2}

Note $\operatorname{nullity}(D)=1$ (constants) and $\operatorname{rank}(D)=3$ , confirming $4 = 3+1$ by Rank–Nullity [6].

⚠️ Common mistake

"Linear" does not mean "graph is a line." A function like $f(x) = mx + b$ with $b \neq 0$ is not a linear transformation — it fails $f(0) = 0$ . Such maps are called affine. Every linear transformation must send $\mathbf{0}$ to $\mathbf{0}$ , because $T(\mathbf{0}) = T(0\cdot\mathbf{0}) = 0\cdot T(\mathbf{0}) = \mathbf{0}$ by axiom [2]. Always check this first as a quick necessary test.

CounterexampleMaps that look linear but aren't

$T(x,y) = (x^2, y)$ fails homogeneity: $T(2,0)=(4,0) \neq 2T(1,0)=(2,0)$ .
$T(x) = |x|$ fails homogeneity for negative scalars: $T(-1\cdot 1)=1 \neq -1\cdot T(1)=-1$ .
$T(x,y) = (x+1, y)$ fails because $T(\mathbf{0}) = (1,0) \neq \mathbf{0}$ .
$T(x,y) = (xy, 0)$ fails additivity: $T((1,0)+(0,1)) = T(1,1) = (1,0)$ , but $T(1,0)+T(0,1)=(0,0)$ .

Conversely, $T(x,y) = (3x - y,\, 2y,\, x+y)$ is linear (each output coordinate is a homogeneous degree-1 polynomial with no constant term).

connection

Linear transformations sit at the hub of linear algebra. They are the morphisms in the category of vector spaces, connecting to Matrix Algebra (the coordinate representation) and Vector Spaces (the objects they act between). The study of which vectors a map merely stretches leads to Eigenvalues and Eigenvectors, and the search for the "best basis" yields Diagonalization and the Singular Value Decomposition. The kernel and image link to Systems of Linear Equations (solving $A\mathbf{x}=\mathbf{b}$ is asking whether $\mathbf{b} \in \operatorname{im}(T)$ ). The set of all linear maps $V \to \mathbb{F}$ forms the Dual Space, and the algebra of operators $V \to V$ generalizes to Group Representation Theory and Functional Analysis in infinite dimensions.

🔬 Where this appears

⏳ History

- 1750: Cramer publishes Cramer's Rule, implicitly treating systems of linear equations as a unified object.

- 1844: Hermann Grassmann's Ausdehnungslehre introduces abstract

n

-dimensional vector spaces and linear independence, far ahead of its time.

- 1858: Arthur Cayley's Memoir on the Theory of Matrices formally defines matrix operations and links them to linear substitutions.

- 1878: Frobenius develops the theory of rank, canonical forms, and the connection between matrices and bilinear forms.

- 1888: Giuseppe Peano gives the first fully axiomatic definition of a vector space and of linear maps in modern abstract form.

- 1900s: Hilbert and Schmidt extend linear operator theory to infinite-dimensional spaces, founding functional analysis.

- 1918–1930s: Emmy Noether and the German school recast linear algebra in coordinate-free, module-theoretic language.

- 1936–1948: Development of numerical linear algebra and (with Turing, von Neumann) practical matrix computation for early computers.

- 1965: Cooley and Tukey's Fast Fourier Transform makes a key linear transform computationally explosive in importance.

open-problem

In finite dimensions the theory is essentially complete: every linear map is a matrix, and structure is captured by rank, eigenvalues, and canonical forms (Jordan, rational, SVD). The genuine subtleties appear in infinite dimensions, where the correspondence breaks down. Not every linear operator is bounded/continuous, the spectrum can be continuous rather than a discrete set of eigenvalues, and unbounded operators (like differentiation on $L^2$ ) require careful domain specification. The invariant subspace problem — does every bounded operator on a separable infinite-dimensional Hilbert space have a nontrivial closed invariant subspace? — remains open and is among the deepest unresolved questions in operator theory. Even in finite dimensions over arbitrary fields, computing canonical forms efficiently and stably (numerically) is an active area, and questions about matrix multiplication complexity (is the exponent $\omega = 2$ ?) remain open.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also