https://teddyrookbook.com/math/linear-algebra/Recent content in Linear Algebra on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/linear-algebra/systems-pictures/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/systems-pictures/See a system as intersecting lines and planes, or as combining columns. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/augmented-matrices/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/augmented-matrices/Swap, scale and add rows without changing the solutions. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/echelon-form/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/echelon-form/Clear entries below each pivot. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/rref/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/rref/Make every pivot 1 with zeros above and below. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/parametric-solutions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/parametric-solutions/Write every solution in terms of the free variables. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/existence-uniqueness/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/existence-uniqueness/Read the answer from the pivot positions. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/linear-combinations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/linear-combinations/Scale and add vectors to reach new ones. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/span/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/span/The span is every linear combination of a set of vectors. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/matrix-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/matrix-equation/Ax is a combination of the columns of A. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/homogeneous-systems/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/homogeneous-systems/Ax = 0 always has the zero solution; the question is whether there are others. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/linear-independence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/linear-independence/No vector in the set is a combination of the others. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/matrix-algebra/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/matrix-algebra/Add, multiply and transpose, minding the order. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/matrix-inverse/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/matrix-inverse/Row-reduce A next to I until A becomes I; then I has become A⁻¹. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/invertible-matrix-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/invertible-matrix-theorem/Many statements about a square matrix are all true or all false together. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/lu-factorization/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/lu-factorization/Record elimination as A = LU. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/linear-transformations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/linear-transformations/T(u + v) = T(u) + T(v) and T(cu) = cT(u). Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/standard-matrix/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/standard-matrix/The columns are the images of the standard basis vectors. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/geometric-maps/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/geometric-maps/Read and build matrices for common plane maps. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/composition-products/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/composition-products/Doing S after T has matrix ST. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/one-to-one-onto/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/one-to-one-onto/Pivots in every column mean one-to-one; in every row mean onto. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/neural-network-layers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/neural-network-layers/A layer computes ReLU(Wx + b); stack two and you can trace the whole network by hand. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/cofactor-expansion/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/cofactor-expansion/Expand along any row or column. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/determinant-properties/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/determinant-properties/Row operations change the determinant in predictable ways; det(AB) = det A det B. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/determinant-geometry/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/determinant-geometry/|det A| is the factor by which A scales area or volume. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/cramers-rule/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/cramers-rule/Solve small systems and invert with determinants. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/light-proofs/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/light-proofs/Prove statements from definitions, one justified step at a time. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/vector-spaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/vector-spaces/A subspace contains 0 and is closed under addition and scaling. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/null-column-spaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/null-column-spaces/Nul A solves Ax = 0; Col A is the span of the columns. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/basis/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/basis/A basis spans and is independent. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/dimension-rank/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/dimension-rank/Every basis has the same size; rank is the dimension of the column space. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/rank-nullity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/rank-nullity/rank + nullity = number of columns. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/change-of-basis/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/change-of-basis/Describe vectors in a new basis with a change-of-coordinates matrix. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/eigen-intro/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/eigen-intro/Av = λv: A only stretches v. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/characteristic-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/characteristic-equation/Solve det(A − λI) = 0. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/eigenspaces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/eigenspaces/Each eigenvalue has a space of eigenvectors. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/diagonalization/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/diagonalization/A = PDP⁻¹ when there are enough independent eigenvectors. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/matrix-powers-markov/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/matrix-powers-markov/Use Aᵏ = PDᵏP⁻¹ and find steady states. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/complex-eigenvalues/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/complex-eigenvalues/Complex eigenvalues signal rotation. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/similar-matrices/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/similar-matrices/Similar matrices describe the same map in different bases. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/orthogonality/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/orthogonality/Vectors are orthogonal when their dot product is 0. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/orthogonal-complements/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/orthogonal-complements/Row space and null space are perpendicular. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/orthogonal-projections/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/orthogonal-projections/Find the closest point in a subspace. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/gram-schmidt-qr/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/gram-schmidt-qr/Turn any basis into an orthonormal one. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/least-squares/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/least-squares/Solve AᵀAx̂ = Aᵀb to fit data. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/orthogonal-matrices/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/orthogonal-matrices/QᵀQ = I: rotations and reflections preserve length. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/spectral-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/spectral-theorem/A symmetric matrix has an orthonormal basis of eigenvectors. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/quadratic-forms/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/quadratic-forms/xᵀAx > 0 for all x ≠ 0 exactly when all eigenvalues are positive. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/svd/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/svd/A = UΣVᵀ for every matrix. Linear Algebra · Math Ladderhttps://teddyrookbook.com/math/linear-algebra/svd-applications/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/linear-algebra/svd-applications/Keep the largest singular values to compress data. Linear Algebra · Math Ladder