APSC 174: Linear Algebra for Engineers

Definition

A linear transformation $T: V \rightarrow W$ between vector spaces $V$ and $W$ is a function that preserves vector addition and scalar multiplication:

$T(u + v) = T(u) + T(v)$ for all $u, v \in V$ (preserves addition)
$T(c \cdot v) = c \cdot T(v)$ for all $v \in V$ and scalars $c$ (preserves scalar multiplication)

Examples

Rotation in $\mathbb{R}^2$

A rotation by angle $\theta$ in the plane is a linear transformation:

T\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}

Differentiation

The differentiation operator $D: P_n \rightarrow P_{n-1}$ defined by $D(p(x)) = p'(x)$ is a linear transformation on the space of polynomials.

Projection

The projection of a vector onto a subspace is a linear transformation.

Matrix Representation

Every linear transformation $T: V \rightarrow W$ between finite-dimensional vector spaces can be represented by a matrix with respect to chosen bases.

If ${v_1, v_2, \ldots, v_n}$ is a basis for $V$ and ${w_1, w_2, \ldots, w_m}$ is a basis for $W$ , then:

Compute $T(v_j)$ for each basis vector $v_j$
Express each $T(v_j)$ as a linear combination of the basis vectors of $W$ :

T(v_j) = a_{1j}w_1 + a_{2j}w_2 + \cdots + a_{mj}w_m

The matrix representation of $T$ is the $m \times n$ matrix:

[T] = \begin{bmatrix} 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{bmatrix}

Properties

Kernel and Image

The kernel (or null space) of $T$ , denoted $\ker(T)$ or $\text{Null}(T)$ , is the set of all vectors in $V$ that map to the zero vector in $W$ :

\ker(T) = \{v \in V : T(v) = 0\}

The image (or range) of $T$ , denoted $\text{Im}(T)$ or $\text{Range}(T)$ , is the set of all vectors in $W$ that are the image of some vector in $V$ :

\text{Im}(T) = \{T(v) : v \in V\}

Rank and Nullity

The rank of $T$ is the dimension of its image: $\text{rank}(T) = \dim(\text{Im}(T))$
The nullity of $T$ is the dimension of its kernel: $\text{nullity}(T) = \dim(\ker(T))$

Rank-Nullity Theorem

For a linear transformation $T: V \rightarrow W$ where $V$ is finite-dimensional:

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

This fundamental theorem connects the dimension of the domain, the dimension of the image, and the dimension of the kernel.

Composition of Linear Transformations

If $S: U \rightarrow V$ and $T: V \rightarrow W$ are linear transformations, then their composition $T \circ S: U \rightarrow W$ defined by $(T \circ S)(u) = T(S(u))$ is also a linear transformation.

In terms of matrix representations, if $[S]$ is the matrix of $S$ and $[T]$ is the matrix of $T$ , then the matrix of $T \circ S$ is the product $[T][S]$ .

Invertible Linear Transformations

A linear transformation $T: V \rightarrow W$ is invertible if there exists a linear transformation $T^{-1}: W \rightarrow V$ such that $T^{-1} \circ T = I_V$ and $T \circ T^{-1} = I_W$ , where $I_V$ and $I_W$ are the identity transformations on $V$ and $W$ respectively.

A linear transformation is invertible if and only if:

It is injective (one-to-one): $\ker(T) = {0}$
It is surjective (onto): $\text{Im}(T) = W$

For finite-dimensional spaces of the same dimension, these conditions are equivalent.

Applications

Linear transformations are fundamental in:

Computer graphics (rotations, scaling, projections)
Quantum mechanics (operators on state spaces)
Signal processing (Fourier transforms)
Machine learning (linear models, dimensionality reduction)
Differential equations (solving systems of linear ODEs)

Linear Transformations