Key Concepts of Linear Transformations

Why This Matters

Linear transformations are the backbone of linear algebra—they're the functions that actually do something to vectors while respecting the structure of vector spaces. When you're tested on this material, you're not just being asked to recall definitions. You're being evaluated on whether you understand how transformations behave, why certain properties guarantee invertibility, and how matrix operations connect to geometric intuition. Every concept here—from kernels to eigenvalues—builds toward solving systems, analyzing stability, and working in different coordinate systems.

Don't just memorize that "the kernel is the set of vectors mapping to zero." Know why that matters: it tells you about information loss, injectivity, and the dimension of your solution space. Understand how composition corresponds to matrix multiplication, and when a transformation can be reversed. These connections between algebraic properties and geometric meaning are exactly what exam questions target—especially in proofs and applications.

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Foundational Definitions and Structure

Linear transformations must satisfy two critical properties that preserve the algebraic structure of vector spaces. These properties—additivity and homogeneity—are what distinguish linear maps from arbitrary functions.

Definition of a Linear Transformation

• Preserves vector addition and scalar multiplication—a function $T: V \to W$ is linear if $T(\mathbf{x} + \mathbf{y}) = T(\mathbf{x}) + T(\mathbf{y})$ for all vectors
• Homogeneity condition requires $T(c\mathbf{x}) = cT(\mathbf{x})$ for any scalar $c$, meaning scaling before or after transformation gives the same result
• Zero vector always maps to zero—this is a direct consequence of linearity and serves as a quick check for whether a function could be linear

Matrix Representation of Linear Transformations

• Every linear transformation has a matrix representation once you fix bases for the domain and codomain vector spaces
• Matrix-vector multiplication computes the transformation: if $A$ represents $T$, then $T(\mathbf{x}) = A\mathbf{x}$
• Basis choice matters—the same transformation has different matrix representations in different bases, which is why change of basis becomes important

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Kernel, Image, and Injectivity/Surjectivity

These concepts describe what a transformation does to the vector space—what it collapses, what it reaches, and whether information is preserved or lost. The Rank-Nullity Theorem connects these ideas quantitatively.

Kernel (Null Space) and Image (Range)

• Kernel $\ker(T) = \{\mathbf{v} \in V : T(\mathbf{v}) = \mathbf{0}\}$—represents the transformation's "information loss" and is always a subspace
• Image $\text{Im}(T) = \{T(\mathbf{v}) : \mathbf{v} \in V\}$—the transformation's "reach" in the codomain, also a subspace
• Rank-Nullity Theorem states $\dim(\ker T) + \dim(\text{Im } T) = \dim(V)$, connecting these two fundamental subspaces

One-to-One and Onto Linear Transformations

• Injective (one-to-one) means $\ker(T) = \{\mathbf{0}\}$—no two different inputs produce the same output
• Surjective (onto) means $\text{Im}(T) = W$—every vector in the codomain is reachable
• Bijective transformations are both injective and surjective, which is precisely when an inverse exists

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Composition and Inverses

When you chain transformations together or undo them, the algebra of matrices mirrors the algebra of functions. Matrix multiplication corresponds to function composition, and matrix inversion corresponds to function inversion.

Composition of Linear Transformations

• Composition preserves linearity—if $T_1$ and $T_2$ are linear, then $(T_2 \circ T_1)(\mathbf{x}) = T_2(T_1(\mathbf{x}))$ is also linear
• Matrix product represents composition—if $A$ represents $T_1$ and $B$ represents $T_2$, then $BA$ represents $T_2 \circ T_1$ (note the order!)
• Order matters because matrix multiplication is not commutative; $T_2 \circ T_1 \neq T_1 \circ T_2$ in general

Inverse Linear Transformations

• Inverse undoes the transformation—$T^{-1}(T(\mathbf{x})) = \mathbf{x}$ and $T(T^{-1}(\mathbf{y})) = \mathbf{y}$
• Exists if and only if $T$ is bijective (both injective and surjective), which for square matrices means $\det(A) \neq 0$
• Inverse matrix $A^{-1}$ satisfies $A^{-1}A = AA^{-1} = I$, and can be computed via row reduction or the adjugate formula

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Eigenvalues and Eigenvectors

Eigenvectors reveal the "natural directions" of a transformation—directions that don't rotate, only stretch or compress. This spectral information is crucial for understanding long-term behavior of iterated transformations and solving differential equations.

Eigenvalues and Eigenvectors of Linear Transformations

• Eigenvector definition—a non-zero vector $\mathbf{v}$ satisfying $T(\mathbf{v}) = \lambda \mathbf{v}$, meaning the transformation only scales it
• Eigenvalue $\lambda$ is the scaling factor; found by solving $\det(A - \lambda I) = 0$ (the characteristic equation)
• Applications include stability analysis, diagonalization, solving systems of differential equations, and principal component analysis

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Geometric Transformations

These concrete examples illustrate how abstract linear transformations manifest geometrically. Each type has a characteristic matrix form that you should recognize.

Rotation and Reflection Transformations

• Rotation by angle $\theta$ in $\mathbb{R}^2$ uses matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$—preserves lengths and angles
• Reflection over a line preserves distances but reverses orientation; determinant equals $-1$
• Both are orthogonal transformations—their matrices satisfy $A^T A = I$, meaning $A^{-1} = A^T$

Scaling and Shear Transformations

• Scaling matrices are diagonal—entries determine stretch factors along coordinate axes; $\begin{pmatrix} k_1 & 0 \\ 0 & k_2 \end{pmatrix}$ scales by $k_1$ horizontally and $k_2$ vertically
• Shear matrices are triangular—they distort shapes by sliding layers; $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$ is a horizontal shear
• Shears preserve area (determinant = 1) but change angles; scaling changes area by factor $|k_1 k_2|$

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Change of Basis

Different bases can dramatically simplify how a transformation looks. The goal is often to find a basis of eigenvectors, making the matrix diagonal.

Change of Basis for Linear Transformations

• Same transformation, different matrix—if $P$ is the change of basis matrix, the new representation is $P^{-1}AP$
• Similar matrices represent the same transformation in different bases; they share eigenvalues, determinant, and trace
• Diagonalization occurs when you can find a basis of eigenvectors, yielding $P^{-1}AP = D$ where $D$ is diagonal

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Quick Reference Table

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Self-Check Questions

1. If $T: \mathbb{R}^3 \to \mathbb{R}^3$ has a two-dimensional kernel, what can you conclude about its image dimension and whether $T$ is invertible?

2. Compare and contrast rotation and reflection transformations in $\mathbb{R}^2$: what properties do they share, and how do their determinants differ?

3. Given that $(AB)^{-1} = B^{-1}A^{-1}$, explain why the order reverses by thinking about composition of transformations.

4. A transformation has eigenvalues $\lambda_1 = 2$ and $\lambda_2 = -1$. What happens to vectors along each eigendirection after applying the transformation twice?

5. Why does changing the basis change the matrix representation but not the kernel dimension or eigenvalues? Connect this to the concept of similar matrices.