# Linear Algebra — Vectors, Matrices, Transformations

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Category: Mathematics
Slides: 13
Updated: 2026-05-17T20:55:52.685Z
Tags: catalog, math, linear, algebra

## Summary

Vectors, matrices, transformations. Key sections include: LINEAR ALGEBRA; 02 Vectors; 03 Vector Spaces; 04 Matrices; 05 Matrix Multiplication; 06 Linear Transformations; 07 Determinant; 08 Eigenvalues & Eigenvectors; 09 Solving Ax = b; 10 Decompositions.

## Slide Outline

1. LINEAR ALGEBRA
2. 02 Vectors
3. 03 Vector Spaces
4. 04 Matrices
5. 05 Matrix Multiplication
6. 06 Linear Transformations
7. 07 Determinant
8. 08 Eigenvalues & Eigenvectors
9. 09 Solving Ax = b
10. 10 Decompositions
11. 11 Applications
12. 12 Modern Frontiers
13. 13 Further Reading

## Slide Transcript

### Slide 1: LINEAR
ALGEBRA

- notebook · ch. 1
- Vectors, matrices, transformations.
- A workbook in thirteen pages — the language of straight things.

### Slide 2: 02Vectors

- vectors
- An arrow in space — magnitude and direction. Or, equivalently, an ordered list of numbers: v = (3, 2).
- addition: place head-to-tail, draw the resultant.
- scalar multiplication: cv stretches by c; negative c flips direction.
- Together these two operations are all of linear algebra.
- u + v = (u₁+v₁, u₂+v₂)

### Slide 3: 03Vector Spaces

- vector spaces
- A vector space V is a set closed under linear combinations: take any vectors in it, scale them, add them — you stay inside.
- Basis — a minimal set of vectors whose linear combinations reach every point in V. Like coordinate axes, but you choose them.
- Dimension — the number of vectors in any basis. ℝ² has dim 2; the space of polynomials of degree ≤ 5 has dim 6.
- span{v₁, v₂, …, vₙ} = { c₁v₁ + ⋯ + cₙvₙ : cᵢ ∈ ℝ }
- If the vᵢ are linearly independent, the representation is unique — that's a basis.

### Slide 4: 04Matrices

- matrices
- A matrix is a rectangular array of numbers. We write A ∈ ℝm×n for m rows and n columns.
- Rows index outputs; columns index inputs.
- Each column is the image of a basis vector.
- Square matrices (m=n) act on a space and return to it.
- Matrices are not just bookkeeping — they are the linear maps.
- a₁₁a₁₂a₁₃
- a₂₁a₂₂a₂₃
- a₃₁a₃₂a₃₃
- A 3×3 matrix — nine entries, nine degrees of freedom.

### Slide 5: 05Matrix Multiplication

- composition
- Matrix multiplication looks bizarre until you realize: it is the composition of linear maps.
- (AB)x = A(Bx)
- First apply B, then A. The product AB is the single matrix that does both. Hence it is not commutative — in general AB ≠ BA, just as putting on socks then shoes differs from shoes then socks.
- Entry rule: (AB)ij = Σₖ aᵢₖ bₖⱼ — row of A dotted with column of B.
- Shape rule: (m×k)·(k×n) = m×n. Inner dimensions must match.

### Slide 6: 06Linear Transformations

- transformations
- A linear transformation preserves addition and scaling. Every such map on ℝⁿ is a matrix.
- Stretch
- diag(2, 1) — pulls along an axis.
- Rotate
- angle θ; columns are (cos θ, sin θ) and (−sin θ, cos θ).
- Shear
- (1, 1; 0, 1) — slants the grid.
- Project
- collapses onto a subspace; loses information.
- A shear: grid still parallel, origin fixed.

### Slide 7: 07Determinant

- determinant
- The determinant det(A) is the signed volume scaling factor of the linear map A. A unit cube of volume 1 becomes a parallelepiped of volume |det(A)|.
- det [ a b
- c d ] = ad − bc
- det(A) = 0 means the map collapses dimension — A is singular, has no inverse.
- Negative determinant means the map flips orientation (mirror).
- det(AB) = det(A) · det(B) — volumes multiply.

### Slide 8: 08Eigenvalues & Eigenvectors

- eigen
- An eigenvector v of A is a direction that the map only scales — it does not turn.
- A v = λ v
- The scalar λ is the matching eigenvalue. Eigen-pairs reveal the intrinsic axes of a transformation — the skeleton beneath the cosmetics.
- Found by solving det(A − λI) = 0, the characteristic polynomial.
- v keeps its line; only its length changes.

### Slide 9: 09Solving Ax = b

- systems
- The fundamental equation. Given matrix A and right-hand side b, find the unknown vector x.
- Existence — a solution exists iff b lies in the column space of A.
- Uniqueness — the solution is unique iff the null space of A is trivial: only x = 0 maps to 0.
- rank(A) + nullity(A) = n
- The rank-nullity theorem: every input dimension is either preserved (rank) or crushed (nullity). Linear algebra's conservation law.

### Slide 10: 10Decompositions

- decompositions
- Hard matrices become easy when factored into structured pieces. Three workhorses:
- LU — A = LU. Lower-triangular times upper-triangular. Solves Ax = b in two cheap sweeps. Underlies Gaussian elimination.
- QR — A = QR. Orthogonal Q times upper-triangular R. The engine of least squares.
- SVD — A = UΣVᵀ. Every matrix is a rotation, then a stretch along orthogonal axes (singular values), then another rotation. The single most useful factorization in applied math: PCA, low-rank approximation, pseudo-inverse, latent semantic analysis.

### Slide 11: 11Applications

- applications
- Computer graphics — every rotation, translation, projection, camera transform is a 4×4 matrix. Pixels on screen are the matrix product of geometry and projection.
- Machine learning — data is matrices, weights are matrices, gradients are matrices. A neural network is mostly Wx + b, repeated.
- Physics — quantum mechanics: states are vectors, observables are Hermitian operators, eigenvalues are measured outcomes. Classical mechanics: rigid-body inertia tensors, normal modes, coupled oscillators.
- A discipline indistinguishable from engineering, science, and economics — once you look closely enough.

### Slide 12: 12Modern Frontiers

- modern
- Tensors — multi-index generalizations of matrices. The native data type of deep learning libraries (PyTorch, JAX). A 4D tensor: batch × channel × height × width.
- Kernel methods — implicit infinite-dimensional feature maps via inner products. The kernel trick made SVMs and Gaussian processes practical.
- Numerical linear algebra — randomized SVD, sketching, iterative Krylov methods make billion-dimensional problems tractable.
- Deep learning — transformers are stacks of matrix multiplications wrapped in nonlinearity. Attention is softmax(QKᵀ/√d) V. The bedrock is unchanged.

### Slide 13: 13Further Reading

- references
- Sheldon Axler — Linear Algebra Done Right. Determinant-free, eigenvalue-first.
- Gilbert Strang — Introduction to Linear Algebra; MIT 18.06 lectures.
- Trefethen & Bau — Numerical Linear Algebra. The numerical bible.
- 3Blue1Brown — Essence of Linear Algebra (YouTube) — the visual gold standard.
- Eigenvectors, intuitively — Eigenvectors Explained (YouTube).
- Close the notebook. Open it again tomorrow. — fin.


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