Matrix Transformations

Why This Matters

Matrix transformations are the bridge between abstract linear algebra and the physical world—they describe how vectors move, stretch, rotate, and project in space. When you're solving systems of differential equations, analyzing stability, or working with change-of-basis problems, you're fundamentally working with transformations. The exam will test whether you understand why a transformation behaves the way it does, not just whether you can multiply matrices.

You're being tested on your ability to recognize transformation types from their matrices, predict geometric effects, compose transformations correctly, and connect eigenvalue analysis to transformation behavior. Don't just memorize the matrix forms—know what each transformation does to the standard basis vectors, how transformations combine, and when a transformation is invertible. Master these concepts, and you'll handle everything from geometric problems to differential equation systems with confidence.

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Transformations That Preserve Structure

These transformations maintain key geometric properties like distances, angles, or orientation. Understanding what each transformation preserves versus changes is crucial for exam questions.

Identity Transformation

• Leaves all vectors unchanged—the "do nothing" transformation that maps every vector to itself
• Represented by the identity matrix $I$, where $I\mathbf{v} = \mathbf{v}$ for all vectors $\mathbf{v}$
• Serves as the multiplicative identity for matrix composition; essential baseline for understanding inverse transformations

Rotation

• Rotates vectors around the origin by angle $\theta$ while preserving distances and angles between vectors
• 2D rotation matrix: $R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}$—orthogonal with determinant 1
• Eigenvalues are complex ($e^{\pm i\theta}$), which explains why rotation has no real eigenvectors except at $\theta = 0$ or $\pi$

Reflection

• Flips vectors across a line or plane, reversing orientation while preserving distances
• Reflection matrix is orthogonal with determinant $-1$—this sign change indicates orientation reversal
• Has eigenvalues $1$ and $-1$, corresponding to vectors parallel and perpendicular to the reflection axis

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Transformations That Change Size or Shape

These transformations alter geometric properties like length, area, or angles. Pay attention to how the determinant relates to area/volume scaling.

Scaling

• Stretches or compresses vectors along coordinate axes by factors $s_x, s_y, \ldots$
• Diagonal scaling matrix: $S = \begin{bmatrix} s_x & 0 \\ 0 & s_y \end{bmatrix}$—eigenvalues equal the scaling factors
• Determinant equals $s_x \cdot s_y$, representing the area scale factor; uniform scaling ($s_x = s_y$) preserves angles

Shear

• Distorts shape by sliding layers parallel to an axis while keeping one direction fixed
• Shear matrix example: $\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$ shears horizontally by factor $k$—parallelograms become slanted
• Determinant equals 1 (area preserved), but angles and distances change; eigenvalue is 1 with algebraic multiplicity 2

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Transformations That Reduce Dimension

Projections collapse space onto a subspace, making them essential for least-squares problems and understanding rank.

Projection

• Maps vectors onto a subspace (line, plane, etc.) by finding the closest point in that subspace
• Projection matrices satisfy $P^2 = P$ (idempotent)—applying the projection twice gives the same result
• Eigenvalues are only 0 and 1; vectors in the target subspace have eigenvalue 1, vectors in the orthogonal complement have eigenvalue 0

Translation (in Homogeneous Coordinates)

• Shifts all points by a fixed vector without rotation or scaling—not a linear transformation in standard coordinates
• Requires homogeneous coordinates: $\begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix}$ embeds translation into matrix multiplication
• Essential for computer graphics where combining translation with other transformations requires a unified matrix framework

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Combining and Analyzing Transformations

These concepts let you work with transformations as a system—composing them, reversing them, and understanding their fundamental behavior.

Composition of Transformations

• Multiply matrices right-to-left: $T_2 T_1 \mathbf{v}$ applies $T_1$ first, then $T_2$—order matters!
• Non-commutativity is critical: rotation then scaling ≠ scaling then rotation in general
• Determinants multiply: $\det(AB) = \det(A)\det(B)$, so composed area scale factors combine multiplicatively

Inverse Transformations

• Reverses the original transformation: $T^{-1}T = TT^{-1} = I$
• Exists only when $\det(T) \neq 0$—singular matrices destroy information and can't be undone
• For orthogonal matrices (rotations, reflections), $T^{-1} = T^T$, making inversion computationally simple

Eigenvalue Decomposition

• Factors a diagonalizable matrix as $A = PDP^{-1}$, where $D$ contains eigenvalues and $P$ contains eigenvectors
• Reveals transformation geometry: eigenvalues show scaling along eigenvector directions; complex eigenvalues indicate rotation
• Powers become trivial: $A^n = PD^nP^{-1}$, essential for solving systems like $\mathbf{x}' = A\mathbf{x}$ in differential equations

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Quick Reference Table

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Self-Check Questions

1. Which two transformations are orthogonal (preserve distances), and how can you distinguish them using the determinant?

2. A matrix satisfies $P^2 = P$ but $P \neq I$. What type of transformation is this, and what are its possible eigenvalues?

3. Compare and contrast scaling and shear: which preserves area, which preserves axis directions, and how do their eigenvalue structures differ?

4. If you apply rotation by $\theta$ followed by rotation by $\phi$, what single transformation results? Does the order matter for rotations about the same axis?

5. (FRQ-style) Given a transformation matrix with complex eigenvalues $\lambda = a \pm bi$ where $a^2 + b^2 = 1$, explain geometrically what this transformation does to vectors and why it has no real eigenvectors.