Types of Linear Transformations

Why This Matters

Linear transformations are the heart of abstract linear algebra—they're the structure-preserving maps that let us understand how vector spaces relate to one another. When you're tested on this material, you're not just being asked to recall matrix forms. You're being evaluated on whether you understand what properties each transformation preserves, how transformations compose, and why certain transformations are invertible while others aren't. These concepts connect directly to eigentheory, dimension theorems, and the classification of operators you'll encounter throughout the course.

The key insight is that linear transformations fall into natural families based on what geometric or algebraic properties they preserve: some keep lengths intact, others maintain orientation, and some collapse dimensions entirely. Don't just memorize the matrix representations—know what each transformation does to the kernel and image, whether it's invertible, and how it behaves under composition. That conceptual understanding is what separates a 5 from a 3 on transformation questions.

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Isometries: Transformations That Preserve Distance

These transformations keep vectors the same "size"—they preserve norms and, consequently, inner products. Geometrically, they're the rigid motions that don't stretch or compress space.

Rotation Transformation

• Rotates vectors around the origin by angle $\theta$, with the standard 2D matrix $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$
• Preserves both length and angles—this makes rotations orthogonal transformations with determinant $+1$
• Always invertible—the inverse is simply rotation by $-\theta$, making these crucial for understanding orthogonal groups

Reflection Transformation

• Flips vectors across a line (2D) or hyperplane (3D+), creating a mirror image of the original configuration
• Orthogonal with determinant $-1$—this negative determinant indicates orientation reversal, distinguishing reflections from rotations
• Self-inverse property—applying a reflection twice returns the original vector, so $T^2 = I$

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Scaling Transformations: Changing Magnitude

These transformations alter the size of vectors while keeping certain structural properties intact. They're characterized by diagonal or scalar matrix representations.

Scaling Transformation

• Multiplies vectors by scalar factors along each axis—represented by diagonal matrices with scaling factors $\lambda_1, \lambda_2, \ldots$ on the diagonal
• Invertible if and only if all scaling factors are nonzero—the inverse simply uses reciprocal factors $1/\lambda_i$
• Eigenvectors are the standard basis vectors—each axis scales independently, making these the simplest non-trivial transformations to analyze

Dilation Transformation

• Uniform scaling by a single factor $k$ from the origin—represented by $kI$, where $I$ is the identity matrix
• Every nonzero vector is an eigenvector with eigenvalue $k$—this is the defining property of scalar operators
• Commutes with all linear transformations—since $kI \cdot A = A \cdot kI$ for any matrix $A$, dilations sit in the center of the algebra

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Degenerate Transformations: Collapsing Dimension

These transformations have nontrivial kernels—they send some nonzero vectors to zero, which means they're not invertible and reduce the effective dimension of the image.

Zero Transformation

• Maps every vector to $\mathbf{0}$—the most extreme case of dimension collapse, with $\ker(T) = V$ and $\text{im}(T) = \{\mathbf{0}\}$
• Rank is zero—by the rank-nullity theorem, $\text{nullity} = \dim(V)$, confirming total collapse
• Absorbing element under composition—$T \circ 0 = 0 \circ T = 0$ for any linear transformation $T$

Projection Transformation

• Projects vectors onto a subspace $W$, decomposing $\mathbf{v} = \mathbf{w} + \mathbf{w}^\perp$ and keeping only $\mathbf{w}$
• Idempotent property: $P^2 = P$—once projected, applying $P$ again changes nothing; this is the defining algebraic condition
• Kernel equals the orthogonal complement—for orthogonal projections, $\ker(P) = W^\perp$ and $\text{im}(P) = W$

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Structure-Altering Transformations: Changing Shape

These transformations modify geometric relationships like angles or parallelism while preserving other properties. They reveal how linear maps can distort without collapsing.

Shear Transformation

• Displaces vectors parallel to a fixed direction proportionally to their distance from a fixed line—a 2D shear matrix looks like $\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}$
• Preserves area (determinant = 1) but distorts angles—parallelograms remain parallelograms, but rectangles become slanted
• Not diagonalizable when $k \neq 0$—shears have repeated eigenvalue 1 but only one independent eigenvector, making them defective matrices

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Boundary Cases and Algebraic Identities

These transformations serve as the "building blocks" and identity elements in the algebra of linear maps.

Identity Transformation

• Maps every vector to itself: $I(\mathbf{v}) = \mathbf{v}$—the multiplicative identity in the ring of linear transformations
• Matrix representation has 1s on the diagonal, 0s elsewhere—denoted $I_n$ for $n \times n$ identity
• Unique transformation satisfying $T \circ I = I \circ T = T$ for all linear $T$—essential for defining inverses

Composition of Linear Transformations

• Applying $S$ after $T$ yields $(S \circ T)(\mathbf{v}) = S(T(\mathbf{v}))$—composition is itself linear, with matrix $[S][T]$
• Order matters: composition is not commutative—in general, $S \circ T \neq T \circ S$, though both products are defined for endomorphisms
• Associative property holds—$(R \circ S) \circ T = R \circ (S \circ T)$, which is why matrix multiplication is associative

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A Note on Translation

Translation Transformation

• Shifts all points by a fixed vector $\mathbf{a}$: $T(\mathbf{v}) = \mathbf{v} + \mathbf{a}$
• Not a linear transformation—it fails $T(\mathbf{0}) = \mathbf{0}$ unless $\mathbf{a} = \mathbf{0}$; it's an affine transformation
• Can be linearized using homogeneous coordinates—embedding in higher dimension allows matrix representation, important for computer graphics

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Quick Reference Table

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Self-Check Questions

1. Which two transformations are both orthogonal but differ in their effect on orientation? What algebraic property distinguishes them?

2. A transformation satisfies $T^2 = T$. Name two transformations with this property and explain how their kernels differ.

3. Compare and contrast scaling and dilation: under what condition is a scaling transformation also a dilation? What property do dilations have that general scalings lack?

4. If you compose a rotation by $\theta$ with a rotation by $\phi$, what transformation results? What if you compose two reflections across different lines?

5. An FRQ asks you to identify a transformation that preserves area but is not diagonalizable. Which transformation fits, and why does it fail to be diagonalizable?