Linear Algebra Review

This review covers everything from the Linear Algebra section. Use it to test yourself across all topics before moving on.

• Vectors: $\langle a, b, c \rangle$ with magnitude $\lVert \mathbf{v} \rVert = \sqrt{a^2 + b^2 + c^2}$
• Dot product: $\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3$ (measures alignment, used for projection and lighting)
• Cross product: $\mathbf{u} \times \mathbf{v}$ gives a perpendicular vector (used for normals and area)

• Addition/subtraction require matching dimensions
• Multiplication: row-by-column dot products, result is $m \times p$ from $(m \times n)(n \times p)$
• Not commutative: $AB \neq BA$ in general
• Determinant: $ad - bc$ for 2x2, tells you invertibility and area scaling
• Inverse: $A^{-1}$ exists only when $\det(A) \neq 0$

• Write as augmented matrix, apply row operations
• REF: staircase of zeros below pivots
• RREF: pivots are 1, zeros above and below
• Three outcomes: unique solution, no solution, infinitely many

• Linearly independent: no vector is a combination of the others
• Basis: independent set that spans the space
• Dimension: number of vectors in any basis

• $A\mathbf{v} = \lambda\mathbf{v}$: eigenvectors are scaled, not rotated
• Find eigenvalues via $\det(A - \lambda I) = 0$
• Diagonalization: $A = PDP^{-1}$ makes $A^n = PD^nP^{-1}$ trivial

• MVP pipeline: Projection $\times$ View $\times$ Model $\times$ vertex
• Lighting: brightness $= \mathbf{n} \cdot \mathbf{L}$
• Collision: projection removes wall-normal component for sliding

Example 1: Vector Projection

Project $\mathbf{u} = \langle 5, 12 \rangle$ onto $\mathbf{v} = \langle 3, 4 \rangle$.

$\mathbf{u} \cdot \mathbf{v} = (5)(3) + (12)(4) = 15 + 48 = 63$

$\lVert \mathbf{v} \rVert^2 = 9 + 16 = 25$

$$\text{proj}_{\mathbf{v}} \mathbf{u} = \frac{63}{25} \langle 3, 4 \rangle = \langle 7.56, 10.08 \rangle$$

Example 2: Matrix Multiplication

$$\begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -2 \\ 3 & 0 \end{bmatrix}$$

This scales by (2, 3) then rotates 90°. Order matters: rotating first then scaling gives a different result.

Example 3: 3x3 Gaussian Elimination

$$\begin{cases} x + y + z = 6 \\ 2x + 3y + z = 14 \\ x - y + 2z = 2 \end{cases}$$

After row reduction: $x = 4$, $y = 2$, $z = 0$.

Example 4: Eigenvalues

$$A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$$

$\det(A - \lambda I) = (4-\lambda)(3-\lambda) - 2 = \lambda^2 - 7\lambda + 10 = (\lambda - 5)(\lambda - 2) = 0$

Eigenvalues: $\lambda_1 = 5$, $\lambda_2 = 2$.

Example 5: Determinant and Invertibility

$$A = \begin{bmatrix} 6 & 3 \\ 4 & 2 \end{bmatrix}$$

$\det(A) = (6)(2) - (3)(4) = 12 - 12 = 0$

$A$ is singular. No inverse exists. The rows are linearly dependent (row 1 = 1.5 $\times$ row 2).

• Practice matrix multiplication and Gaussian elimination until they feel automatic
• Visualize vectors and transformations whenever possible
• The key insight: matrices represent linear transformations
• Connect concepts: dot product leads to projection leads to lighting; eigenvalues lead to diagonalization leads to efficient computation
• When stuck on a problem, ask: “What does this look like geometrically?”