Linear Maps: Isomorphisms and Homomorphisms

Structure-Preserving Maps

Given two vector spaces $V$ and $W$, a function $h: V \to W$ is a homomorphism (or linear map) if it preserves the vector space structure:

$$h(\vec{v}_1 + \vec{v}_2) = h(\vec{v}_1) + h(\vec{v}_2) \qquad h(c\vec{v}) = c\,h(\vec{v})$$

Equivalently, for any scalars $a, b$ and vectors $\vec{v}_1, \vec{v}_2$:

$$h(a\vec{v}_1 + b\vec{v}_2) = a\,h(\vec{v}_1) + b\,h(\vec{v}_2)$$

A linear map is completely determined by where it sends each basis vector. If $B = \{\vec{\beta}_1,\ldots,\vec{\beta}_n\}$ is a basis for $V$ and you specify $h(\vec{\beta}_i) \in W$ freely, then $h$ extends uniquely to all of $V$ by linearity:

$$h\!\left(\sum c_i \vec{\beta}_i\right) = \sum c_i\, h(\vec{\beta}_i)$$

This is why matrices exist: a matrix is just a compact encoding of where a linear map sends each standard basis vector.

Examples

Zero map: $\vec{v} \mapsto \vec{0}$ for all $\vec{v}$. Trivially linear.

Identity: $\text{id}_V: \vec{v} \mapsto \vec{v}$. Linear.

Rotation in $\mathbb{R}^2$: $R_\theta(x, y) = (x\cos\theta - y\sin\theta,\; x\sin\theta + y\cos\theta)$. Linear — it preserves addition and scaling.

Projection: $\pi: \mathbb{R}^3 \to \mathbb{R}^2$, $(x,y,z) \mapsto (x,y)$. Linear.

Differentiation: $D: \mathcal{P}_n \to \mathcal{P}_{n-1}$, $p(x) \mapsto p'(x)$. Linear — the derivative of a sum is the sum of derivatives.

Not linear: $f(x) = x + 1$ is not linear because $f(0) = 1 eq 0$ (a linear map must send $\vec{0}$ to $\vec{0}$).

Range Space and Null Space

For $h: V \to W$:

• The range space $\mathcal{R}(h) = \{h(\vec{v}) : \vec{v} \in V\}$ is the image of $h$ — the set of all vectors $W$ that $h$ can produce. It is a subspace of $W$.
• The null space $\mathcal{N}(h) = \{\vec{v} \in V : h(\vec{v}) = \vec{0}\}$ is the kernel — the set of all inputs that map to zero. It is a subspace of $V$.

These subspaces measure what the map does and what it kills. The rank-nullity theorem applies:

$$\dim(\mathcal{R}(h)) + \dim(\mathcal{N}(h)) = \dim(V)$$

Isomorphisms

An isomorphism is a linear map $f: V \to W$ that is also a bijection (one-to-one and onto). When an isomorphism exists, $V$ and $W$ are isomorphic: $V \cong W$.

Isomorphic spaces are structurally identical — they differ only in the labeling of their elements. No linear-algebraic property can distinguish them.

The fundamental theorem: Every finite-dimensional real vector space of dimension $n$ is isomorphic to $\mathbb{R}^n$.

This theorem is why $\mathbb{R}^n$ is sufficient to study all finite-dimensional vector spaces. The space of $3 \times 3$ matrices ($\dim = 9$) is isomorphic to $\mathbb{R}^9$; $\mathcal{P}_4$ ($\dim = 5$) is isomorphic to $\mathbb{R}^5$.

Characterizing Isomorphisms

A linear map $f: V \to W$ is an isomorphism iff:

• Injective (one-to-one): $\mathcal{N}(f) = \{\vec{0}\}$ — no two vectors map to the same output
• Surjective (onto): $\mathcal{R}(f) = W$ — every output is achieved

For maps between spaces of the same dimension, injectivity and surjectivity are equivalent — checking either one suffices.

Transformations

A transformation is a linear map from a space to itself: $t: V \to V$. Every $n \times n$ matrix represents a transformation of $\mathbb{R}^n$.

Transformations can be composed: $(t \circ s)(\vec{v}) = t(s(\vec{v}))$, corresponding to matrix multiplication. Powers of a transformation $t^k = t \circ \cdots \circ t$ correspond to matrix powers $A^k$.

A transformation $t$ is invertible (an isomorphism from $V$ to $V$) iff $\mathcal{N}(t) = \{\vec{0}\}$, iff $\det(t) eq 0$ — connecting back to the Invertible Matrix Theorem.