Linear Maps

Definition

A function $f: V \to W$ between vector spaces is linear if:

$f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})$ $f(\lambda\mathbf{u}) = \lambda f(\mathbf{u})$

for all $\mathbf{u}, \mathbf{v} \in V$ and $\lambda \in K$.

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Matrix Representation

Every linear map $f: \mathbb{R}^n \to \mathbb{R}^m$ can be written:

$f(\mathbf{x}) = A\mathbf{x}$

where $A$ is an $m \times n$ matrix.

The $j$th column of $A$ is the image of the $j$th basis vector: $A\mathbf{e}_j$.

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Orthogonal Matrices

A matrix $O$ is orthogonal if:

$O^T O = I$

Properties:

• $O^{-1} = O^T$
• Columns form an orthonormal basis
• $\det(O) = \pm 1$
• Preserves length: $|O\mathbf{x}| = |\mathbf{x}|$
• Preserves dot product: $(O\mathbf{u}) \cdot (O\mathbf{v}) = \mathbf{u} \cdot \mathbf{v}$

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Rotation Matrices

2D rotation by angle $\theta$:

$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$

Properties:

• $R(\theta)R(\phi) = R(\theta + \phi)$
• $R(\theta)^{-1} = R(-\theta)$

3D rotations about coordinate axes:

$R_x(\theta) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta \\ 0 & \sin\theta & \cos\theta \end{pmatrix}$

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Reflection Matrices

2D reflection across line at angle $\theta$ to $x$-axis:

$M(\theta) = \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix}$

$\det(M) = -1$ (orientation-reversing)

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Geometric Transformations

Transformation	Matrix type
Rotation	Orthogonal, $\det = +1$
Reflection	Orthogonal, $\det = -1$
Scaling	Diagonal (or isotropic)
Shear	Upper/lower triangular with 1s on diagonal
Projection	$A^2 = A$ (idempotent)

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Determinant and Volume

For linear map with matrix $A$:

$|\det(A)| = \frac{\text{volume of image}}{\text{volume of original}}$

• $\det > 0$: orientation preserved
• $\det < 0$: orientation reversed
• $\det = 0$: dimension collapses (singular)

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Kernel and Image of Linear Map

For $f: V \to W$:

Kernel: $\ker(f) = \{\mathbf{v} \in V : f(\mathbf{v}) = \mathbf{0}\}$

Image: $\text{im}(f) = \{f(\mathbf{v}) : \mathbf{v} \in V\}$

Rank-nullity theorem:

$\dim(V) = \dim(\ker(f)) + \dim(\text{im}(f))$

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Change of Basis

If $A$ represents a linear map in basis $\{\mathbf{e}_i\}$, then in basis $\{\mathbf{f}_i\}$ where $\mathbf{f}_j = \sum P_{ij}\mathbf{e}_i$:

$A' = P^{-1}AP$

The same linear map has different matrix representations in different bases.

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Why Linear Maps Matter

• Unified treatment of rotations, scaling, projections
• Essential for computer graphics
• Foundation for eigentheory (best basis for a linear map)
• Quantum mechanics: observables are linear operators

Linear maps preserve vector space structure — this is the essence of linearity.