shipslides
Mathematics13 slides0 views

Linear Algebra — Vectors, Matrices, Transformations

Vectors, matrices, transformations.

StandaloneDownload
Sandboxed deck
Open raw
Shared with ShipslidesCreate your own deck →

About this HTML presentation

This Shipslides page presents Linear Algebra — Vectors, Matrices, Transformations as an interactive HTML presentation deck in the Mathematics catalog with 13 slides. The share page keeps the uploaded deck sandboxed while exposing readable context, topics, and a slide outline for viewers and search engines.

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.

Key sections

  • 01LINEAR ALGEBRA
  • 0202 Vectors
  • 0303 Vector Spaces
  • 0404 Matrices
  • 0505 Matrix Multiplication
  • 0606 Linear Transformations
  • 0707 Determinant
  • 0808 Eigenvalues & Eigenvectors
  • 0909 Solving Ax = b
  • 1010 Decompositions
  • 1111 Applications
  • 1212 Modern Frontiers
  • 1313 Further Reading

Topics covered

catalogmathlinearalgebra

Related decks

Mathematics30 slides

Linear Algebra

Mathematics13 slides

Calculus — The Mathematics of Change

Mathematics13 slides

Mathematical Cryptography

Mathematics14 slides

Game Theory — Strategy when others strategize too

Slide outline
  1. 01LINEAR ALGEBRA
  2. 0202 Vectors
  3. 0303 Vector Spaces
  4. 0404 Matrices
  5. 0505 Matrix Multiplication
  6. 0606 Linear Transformations
  7. 0707 Determinant
  8. 0808 Eigenvalues & Eigenvectors
  9. 0909 Solving Ax = b
  10. 1010 Decompositions
  11. 1111 Applications
  12. 1212 Modern Frontiers
  13. 1313 Further Reading
Page data
Canonical
https://shipslides.com/d/catalog-math-linear-algebra
Category
Mathematics
Size
31.2 KB
Updated
2026-05-17
LLM text
https://shipslides.com/d/catalog-math-linear-algebra/llms.txt

Presentation Transcript

Detailed slide-by-slide text content extracted from this presentation.

Slide 01

LINEAR ALGEBRA

  • notebook · ch. 1
  • Vectors, matrices, transformations.
  • A workbook in thirteen pages — the language of straight things.
Slide 02

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 03

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 04

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 05

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 06

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 07

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 08

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 09

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.
Remove this deck