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Linear Algebra | Math Infinitum

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Linear Algebra

Mathematics · Course 01

Linear Algebra

A language for space, movement, and transformation.

Start with an arrow. Build a way to think about vectors, coordinates, and the spaces they describe.

A x = b

A v = λ v

det (A)

u \cdot v

(a c b d)

span (v_{1}, \dots, v_{n})

A = P D P^{- 1}

rank (A)

∥ v ∥

e_{1}, e_{2}, e_{3}

Course map

11 chapters

01VectorsDirection and magnitude. One idea, two ways to see it.
02Vectors and linear systemsFrom a list of numbers to linear combinations, and from a system of equations to the question of whether it can be solved at all.
03Matrices and transformationsFrom a system’s coefficient vectors to a matrix: its special shapes, its arithmetic, the map it performs, products, inverses and the transpose.
04DeterminantsOne number from a square matrix: how to compute it at every size, what it measures, how row operations change it, and how cofactors break it into smaller ones.
05Inverse matricesWhen a square matrix can be undone, and how to write its inverse down: the 2 by 2 formula, the rules inverses obey, the adjugate and Cramer’s rule.

Systems and Gaussian eliminationSolving any linear system by hand: the augmented matrix, echelon forms and pivots, free variables, the rank, and inverses found by row reduction.

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07Vector spaces and basesWhat a vector space is, beyond lists of numbers: subspaces, independence, bases, coordinates and dimension, and the rank seen as the size of what a matrix can reach.↗

08Linear transformationsLinear maps between any two vector spaces: pictures of them in the plane, their kernels and images, the rank–nullity theorem, their matrices in any basis, change of basis, and every way of saying that a square matrix is invertible.↗

09Eigenvalues and eigenvectorsThe directions a map only stretches: eigenvalues from the characteristic polynomial, eigenspaces and the two multiplicities, diagonalisation, and what eigenvalues say about powers and the long run.↗

10Orthogonality and least squaresLengths and angles from the dot product and from inner products in general: orthogonal bases and matrices, orthogonal complements, projections, the Gram–Schmidt process, and least squares for systems that have no solution.↗

11Symmetric matrices and the spectral theoremWhere eigenvalues and orthogonality meet: symmetric matrices, their orthonormal bases of eigenvectors, the quadratic forms they describe, and a review of the whole course.↗