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Linear Algebra

Vectors and the matrices that move them. Every operation is shown row by row, and 2×2 and 3×3 matrices are drawn in 3D so you can see what a determinant or an eigenvector actually is.

الدروس المستفادة

Core Determinants 2×2 by ad − bc, larger by cofactor expansion — and what the number means. det [[1,2],[3,4]] Core Matrix inverse The 2×2 formula and Gauss–Jordan elimination for the rest. inverse of [[1,2],[3,4]] Advanced Eigenvalues and eigenvectors The characteristic polynomial and the directions a matrix only stretches. eigenvalues of [[2,1],[1,2]] Introductory Matrix multiplication Rows times columns, entry by entry. [[1,2],[3,4]] * [[5,6],[7,8]] Core Row reduction Gaussian elimination, rank and solving Ax = b. rref [[1,2,3],[4,5,6],[7,8,9]] Core Vector spaces, span and linear independence The axioms, spanning sets, independence, basis and dimension. rank of [[1,2],[2,4]] Core Orthogonality, projections and least squares Dot products, orthonormal bases, projection onto a subspace, fitting a line. inverse of [[2,1],[1,1]] Core Diagonalisation and matrix powers A = PDP⁻¹, computing Aⁿ, and the matrices that cannot be diagonalised. eigenvalues of [[4,1],[2,3]] Core Linear transformations and change of basis Matrices as maps, kernel and image, and the same map in different coordinates. [[0,-1],[1,0]] * [[1],[0]]

Chapters from Beezer, A First Course in Linear Algebra

Every section of the book, condensed into a lesson with its own practice problems.

Chapters from OpenStax Precalculus 2e

Every section of the book, condensed into a lesson with its own practice problems.

9. Systems of Equations and Inequalities

Introduction to Systems of Equations and Inequalities Systems of Linear Equations: Two Variables Systems of Linear Equations: Three Variables Systems of Nonlinear Equations and Inequalities: Two Variables Partial Fractions Matrices and Matrix Operations Solving Systems with Gaussian Elimination Solving Systems with Inverses Solving Systems with Cramer's Rule

Symbols used here

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}

matrix
A rectangular array of numbers; a linear map.

\det A,\ |A|

determinant
Scaling factor of area/volume under A; zero means singular.

A^{-1},\ A^{T}

inverse, transpose
The matrix that undoes A; A with rows and columns swapped.

\lambda

lambda (eigenvalue)
The factor by which an eigenvector is stretched: Av = λv.

\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|

dot product, norm
Σ u_i v_i; the length of v, √(v·v).

Questions people ask

What does a determinant mean geometrically?

It is the factor by which the matrix scales area (2×2) or volume (3×3), with a negative sign if orientation flips. Zero means the matrix flattens space and cannot be undone.

What is an eigenvector?

A direction the matrix does not turn — it only stretches it by the eigenvalue. Along eigenvectors a complicated matrix acts like multiplication by a number.

Why is matrix multiplication not commutative?

Because a matrix is a transformation and AB means "do B, then A". Rotating then reflecting is not the same as reflecting then rotating.

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＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry π₁ Algebraic Topology → Category Theory ∞ Frontiers

⊞ Linear Algebra

الدروس المستفادة

Chapters from Beezer, A First Course in Linear Algebra

1. Systems of Linear Equations

2. Vectors

3. Matrices

4. Vector Spaces

5. Determinants

6. Eigenvalues

7. Linear Transformations

8. Representations