[{"data":1,"prerenderedAt":1279},["ShallowReactive",2],{"subject:linear-algebra":3,"course-wordcounts":75,"nav:linear-algebra":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F02.linear-algebra\u002Findex.md","Linear Algebra","Vectors, matrices, and the geometry of linear maps — solving systems,\nfactoring matrices, and finding the eigenvectors and singular values that\nreveal what a transformation really does.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Two threads run through the subject: solving linear systems, and understanding\nthe linear maps behind them. The sequence starts with vectors and matrix\narithmetic, then treats a matrix as a transformation of space — the picture that\nmakes everything else intuitive. Gaussian elimination and the LU factorization\nsettle when a system is solvable and how to solve it; vector spaces, independence,\nspan, basis, and rank organize the solution sets; determinants track volume and\ninvertibility. From there the notes build the geometry — inner products,\northogonality, and projection — and the two great factorizations: the eigenvalue\ndecomposition, which finds the directions a map only scales, and the singular\nvalue decomposition, which does the same for every matrix at once. A closing\nmodule looks at how these computations actually behave in floating-point\narithmetic. These notes follow Lay & McDonald.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,28,30,34,36,40,42,46,48,52,54],{"p":20},"Linear algebra is the study of \u003Cstrong>linear maps\u003C\u002Fstrong> and the spaces\nthey act on. Vectors are the objects; a matrix is the map — and almost\nevery question in the subject is really a question about what that map\ndoes to space.\n",{"fig":22,"n":23,"caption":24,"large":25},"la-transform","001","A matrix as a transformation: the square grid shears into parallelograms,\ncarrying the basis vectors to its columns.\n",true,{"p":27},"Read a matrix as a \u003Cem>verb\u003C\u002Fem>, not a table. Its columns record exactly\nwhere the basis vectors land, and that alone fixes the fate of every other\nvector — the grid deforms with them.\n",{"p":29},"The first hard question is when a system \u003Cem>Ax = b\u003C\u002Fem> has a solution at\nall. Gaussian elimination answers it mechanically: clear each column below\nits pivot and the matrix steps into row-echelon form, where the solutions\ncan simply be read off.\n",{"fig":31,"n":32,"caption":33},"la-gauss","002","Gaussian elimination: entries below each pivot fall to zero, stepping the\nmatrix to row-echelon form.\n",{"p":35},"Beneath the mechanics sit the \u003Cstrong>vector spaces\u003C\u002Fstrong> that organize\nthe answers. Independence, span, and basis measure how many directions a\nset of vectors really reaches, and dimension counts them.\n",{"fig":37,"n":38,"caption":39},"la-span","003","Span: every scalar combination of two independent vectors sweeps out the\nwhole plane they generate.\n",{"p":41},"Some directions survive a transformation untouched in orientation. An\n\u003Cstrong>eigenvector\u003C\u002Fstrong> is carried straight back onto its own line,\nmerely stretched by an eigenvalue — the axes along which the map acts most\nsimply.\n",{"fig":43,"n":44,"caption":45},"la-eigen","004","Eigenvectors: a generic vector is rotated, but these special directions are\nonly scaled by \u003Cem>λ\u003C\u002Fem>.\n",{"p":47},"Orthogonality turns geometry into computation. Projecting a vector onto a\nsubspace drops a perpendicular to the closest point in it — the idea behind\nleast squares, orthonormal bases, and the singular value decomposition.\n",{"fig":49,"n":50,"caption":51},"la-projection","005","Orthogonal projection: the foot p is the closest point on the line to b,\nmet at a right angle.\n",{"p":53},"Determinants measure how a map scales volume and whether it is invertible;\nan eigenbasis diagonalizes it; and the SVD factors \u003Cem>any\u003C\u002Fem> matrix into\na rotation, a set of stretches, and another rotation.\n",{"p":55},"The payoff is a single language for systems of equations, geometry, and\ndata — and a set of factorizations that make each of them computable,\neven in floating point.\n","math","Linear algebra is the study of linear equations and the spaces their\nsolutions live in. Row reduction settles when a system is solvable; vector\nspaces and bases organize the answers; determinants, eigenvalues, and\northogonality expose the structure of a matrix; and the SVD ties all of it\ntogether. These notes follow Lay & McDonald, with a closing module on how\nthe computations behave in floating point.\n",false,"md",{},"\u002Flinear-algebra",[],"---\ntitle: Linear Algebra\nstatus: available\ncategory: math\nblurb: |\n  Vectors, matrices, and the geometry of linear maps — solving systems,\n  factoring matrices, and finding the eigenvectors and singular values that\n  reveal what a transformation really does.\ndescription: |\n  Linear algebra is the study of linear equations and the spaces their\n  solutions live in. Row reduction settles when a system is solvable; vector\n  spaces and bases organize the answers; determinants, eigenvalues, and\n  orthogonality expose the structure of a matrix; and the SVD ties all of it\n  together. These notes follow Lay & McDonald, with a closing module on how\n  the computations behave in floating point.\nbrief:\n  - p: |\n      Linear algebra is the study of \u003Cstrong>linear maps\u003C\u002Fstrong> and the spaces\n      they act on. Vectors are the objects; a matrix is the map — and almost\n      every question in the subject is really a question about what that map\n      does to space.\n  - fig: la-transform\n    n: \"001\"\n    caption: |\n      A matrix as a transformation: the square grid shears into parallelograms,\n      carrying the basis vectors to its columns.\n    large: true\n  - p: |\n      Read a matrix as a \u003Cem>verb\u003C\u002Fem>, not a table. Its columns record exactly\n      where the basis vectors land, and that alone fixes the fate of every other\n      vector — the grid deforms with them.\n  - p: |\n      The first hard question is when a system \u003Cem>Ax = b\u003C\u002Fem> has a solution at\n      all. Gaussian elimination answers it mechanically: clear each column below\n      its pivot and the matrix steps into row-echelon form, where the solutions\n      can simply be read off.\n  - fig: la-gauss\n    n: \"002\"\n    caption: |\n      Gaussian elimination: entries below each pivot fall to zero, stepping the\n      matrix to row-echelon form.\n  - p: |\n      Beneath the mechanics sit the \u003Cstrong>vector spaces\u003C\u002Fstrong> that organize\n      the answers. Independence, span, and basis measure how many directions a\n      set of vectors really reaches, and dimension counts them.\n  - fig: la-span\n    n: \"003\"\n    caption: |\n      Span: every scalar combination of two independent vectors sweeps out the\n      whole plane they generate.\n  - p: |\n      Some directions survive a transformation untouched in orientation. An\n      \u003Cstrong>eigenvector\u003C\u002Fstrong> is carried straight back onto its own line,\n      merely stretched by an eigenvalue — the axes along which the map acts most\n      simply.\n  - fig: la-eigen\n    n: \"004\"\n    caption: |\n      Eigenvectors: a generic vector is rotated, but these special directions are\n      only scaled by \u003Cem>λ\u003C\u002Fem>.\n  - p: |\n      Orthogonality turns geometry into computation. Projecting a vector onto a\n      subspace drops a perpendicular to the closest point in it — the idea behind\n      least squares, orthonormal bases, and the singular value decomposition.\n  - fig: la-projection\n    n: \"005\"\n    caption: |\n      Orthogonal projection: the foot p is the closest point on the line to b,\n      met at a right angle.\n  - p: |\n      Determinants measure how a map scales volume and whether it is invertible;\n      an eigenbasis diagonalizes it; and the SVD factors \u003Cem>any\u003C\u002Fem> matrix into\n      a rotation, a set of stretches, and another rotation.\n  - p: |\n      The payoff is a single language for systems of equations, geometry, and\n      data — and a set of factorizations that make each of them computable,\n      even in floating point.\n---\n\nTwo threads run through the subject: solving linear systems, and understanding\nthe linear maps behind them. The sequence starts with vectors and matrix\narithmetic, then treats a matrix as a transformation of space — the picture that\nmakes everything else intuitive. Gaussian elimination and the LU factorization\nsettle when a system is solvable and how to solve it; vector spaces, independence,\nspan, basis, and rank organize the solution sets; determinants track volume and\ninvertibility. From there the notes build the geometry — inner products,\northogonality, and projection — and the two great factorizations: the eigenvalue\ndecomposition, which finds the directions a map only scales, and the singular\nvalue decomposition, which does the same for every matrix at once. A closing\nmodule looks at how these computations actually behave in floating-point\narithmetic. These notes follow Lay & McDonald.\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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Equations in Linear Algebra",1,"linear-systems",[993,998,1003,1009,1015],{"title":994,"path":995,"lessonNumber":990,"topics":996,"summary":997},"Systems of Linear Equations and Row Reduction","\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms",[989],"A linear system is a finite set of linear equations in shared variables. Elementary row operations rewrite it without changing its solution set, and reducing the augmented matrix to echelon form decides both existence and uniqueness. Pivot positions say whether the solution set is empty, a single point, or infinite.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"Vector Equations and the Matrix Equation Ax = b","\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations",[989],"The same linear system reads three equivalent ways: a system of equations, a vector equation asking whether b is a linear combination of fixed vectors, and a matrix equation Ax = b. Ax is the linear combination of A's columns weighted by x, so consistency for a given b means b lies in the span of the columns, and consistency for every b means the columns span all of R^m.\n",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"Solution Sets and Applied Linear Systems","\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications",3,[989],"A homogeneous system Ax = 0 has a solution set that is a span through the origin; a consistent Ax = b has that same span translated by any one particular solution. Parametric vector form writes both explicitly. The structure shows up in applied systems with many solutions: equilibrium prices, balanced chemical reactions, network flows, weight-loss diets, and migration models.\n",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Linear Independence","\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence",4,[989],"A set of vectors is linearly independent when the only linear combination equal to zero is the trivial one; otherwise a dependence relation writes one vector in terms of the others. For the columns of A the question becomes whether Ax = 0 has only the trivial solution — a pivot in every column. Counting pivots settles independence, and any set with more vectors than entries is automatically dependent.\n",{"title":1016,"path":1017,"lessonNumber":1018,"topics":1019,"summary":1020},"Linear Transformations and Their Matrices","\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations",5,[989],"Reading A as an action rather than an array, x maps to Ax is a transformation from R^n to R^m. The ones that preserve addition and scalar multiplication are the linear transformations, and every one is x maps to Ax for a unique standard matrix whose columns are the images of the standard basis vectors. Onto and one-to-one translate into the span and independence of those columns.\n",{"module":1022,"moduleNumber":16,"slug":1023,"lessons":1024},"Matrix Algebra","matrix-algebra",[1025,1030,1035,1040,1045],{"title":1026,"path":1027,"lessonNumber":990,"topics":1028,"summary":1029},"Matrix Operations","\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations",[1022],"Matrices add and scale entrywise, but their product is defined so that multiplication corresponds to composition of linear maps: the columns of AB are A applied to the columns of B. From that requirement follow the row-column rule, the algebra of products (associative and distributive but not commutative), powers, and the transpose.\n",{"title":1031,"path":1032,"lessonNumber":16,"topics":1033,"summary":1034},"The Inverse and the Invertible Matrix Theorem","\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility",[1022],"The inverse of a square matrix is the matrix analogue of a reciprocal, defined by AA⁻¹ = I. A closed form settles the 2×2 case; the Gauss–Jordan algorithm row reduces [A | I] to [I | A⁻¹] in general; and elementary matrices record single row operations. The Invertible Matrix Theorem collects a dozen equivalent conditions for invertibility into one statement.\n",{"title":1036,"path":1037,"lessonNumber":1006,"topics":1038,"summary":1039},"Block Matrices and the LU Factorization","\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu",[1022],"Partitioning a matrix into blocks lets sums, products, and inverses be computed block by block, as if the submatrices were scalars. Block structure also underlies the LU factorization A = LU, which splits solving Ax = b into two fast triangular solves and repays the cost whenever many systems share one coefficient matrix.\n",{"title":1041,"path":1042,"lessonNumber":1012,"topics":1043,"summary":1044},"Subspaces of Rⁿ, Dimension, and Rank","\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank",[1022],"A subspace is a set closed under addition and scalar multiplication. Every matrix carries two: the column space of all attainable outputs Ax, and the null space of all solutions of Ax = 0. A basis measures each with a minimal spanning set, dimension counts it, and the Rank Theorem ties pivots and free variables together as rank + nullity = n.\n",{"title":1046,"path":1047,"lessonNumber":1018,"topics":1048,"summary":1049},"Applications: Leontief Economics and Computer Graphics","\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics",[1022],"The Leontief input–output model balances an economy through (I − C)x = d and expands the inverse as a geometric series in the consumption matrix. Computer graphics moves figures with matrix products, using homogeneous coordinates so that translation and perspective projection become matrix multiplications too.\n",{"module":1051,"moduleNumber":1006,"slug":1052,"lessons":1053},"Determinants","determinants",[1054,1059,1064],{"title":1055,"path":1056,"lessonNumber":990,"topics":1057,"summary":1058},"Introduction to Determinants","\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors",[1051],"The determinant of a square matrix is defined recursively by cofactor expansion: an n-by-n determinant is a signed sum of (n-1)-by-(n-1) determinants built from the first row. The expansion can equally run along any row or down any column, and a triangular matrix has determinant equal to the product of its diagonal.\n",{"title":1060,"path":1061,"lessonNumber":16,"topics":1062,"summary":1063},"Properties of Determinants","\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants",[1051],"Row operations act on the determinant in three predictable ways, and this turns row reduction into a fast algorithm: the determinant is the product of the pivots times a sign for the interchanges. The same properties yield the invertibility test det A is nonzero, the transpose identity, and the multiplicative law det(AB) equals det A times det B.\n",{"title":1065,"path":1066,"lessonNumber":1006,"topics":1067,"summary":1068},"Cramer's Rule, Volume, and Linear Transformations","\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area",[1051],"Cramer's rule writes each unknown of an invertible system as a ratio of determinants, and the same idea gives a closed formula for the inverse through the adjugate. Geometrically the absolute determinant is the area of the parallelogram or the volume of the parallelepiped spanned by the columns, so a linear map scales every region's measure by that factor.\n",{"module":1070,"moduleNumber":1012,"slug":1071,"lessons":1072},"Vector Spaces","vector-spaces",[1073,1078,1083,1088,1093,1098,1104],{"title":1074,"path":1075,"lessonNumber":990,"topics":1076,"summary":1077},"Vector Spaces and Subspaces","\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces",[1070],"A vector space is any set closed under addition and scalar multiplication that obeys ten algebraic axioms. The same axioms that govern arrows in the plane govern polynomials, functions, matrices, and infinite signals, so one theory covers them all. A subspace is a subset that is a vector space in its own right, tested by three conditions, and the span of any set of vectors is the smallest subspace containing them.\n",{"title":1079,"path":1080,"lessonNumber":16,"topics":1081,"summary":1082},"Null Spaces, Column Spaces, and Linear Transformations","\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces",[1070],"Two subspaces sit inside every matrix. The null space collects all solutions of $Ax = 0$ and lives in the domain; the column space collects every attainable $Ax$ and lives in the codomain. One is defined implicitly by a condition, the other explicitly by a spanning set, and the same pair appears for an abstract linear transformation as its kernel and range.\n",{"title":1084,"path":1085,"lessonNumber":1006,"topics":1086,"summary":1087},"Linearly Independent Sets and Bases","\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets",[1070],"A basis is a spanning set with no redundancy: linearly independent and still large enough to reach every vector. The spanning-set theorem shows any spanning set can be trimmed to a basis by discarding dependent vectors, and the pivot columns of a matrix give a basis for its column space. Independence and spanning are defined for abstract spaces exactly as in $\\mathbb{R}^n$.\n",{"title":1089,"path":1090,"lessonNumber":1012,"topics":1091,"summary":1092},"Coordinate Systems","\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems",[1070],"Fixing a basis assigns every vector a unique list of coordinates, turning an abstract space into $\\mathbb{R}^n$. The coordinate mapping is a one-to-one linear transformation onto $\\mathbb{R}^n$ — an isomorphism — so any $n$-dimensional space is indistinguishable from $\\mathbb{R}^n$ as far as vector-space computations go. In $\\mathbb{R}^n$ the change-of-coordinates matrix $P_B$ and its inverse convert between basis coordinates and standard coordinates.\n",{"title":1094,"path":1095,"lessonNumber":1018,"topics":1096,"summary":1097},"The Dimension of a Vector Space and Rank","\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank",[1070],"Every basis of a space has the same number of vectors, and that number is the dimension. Rank is the dimension of the column space, equal to the dimension of the row space and to the number of pivots. The Rank Theorem, rank plus nullity equals the number of columns, ties the four fundamental subspaces of a matrix together and adds six lines to the Invertible Matrix Theorem.\n",{"title":1099,"path":1100,"lessonNumber":1101,"topics":1102,"summary":1103},"Change of Basis","\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis",6,[1070],"Two bases give the same vector two different coordinate vectors, and a single invertible matrix converts between them. Its columns are the coordinate vectors of the old basis expressed in the new one, and its inverse reverses the conversion. In $\\mathbb{R}^n$ the change-of-coordinates matrix between two bases is found by one row reduction.\n",{"title":1105,"path":1106,"lessonNumber":1107,"topics":1108,"summary":1109},"Applications: Difference Equations and Markov Chains","\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov",7,[1070],"The solutions of an nth-order linear difference equation form an $n$-dimensional vector space, so finding $n$ independent solutions gives them all. A Markov chain evolves a probability distribution by repeated multiplication by a stochastic matrix, and a regular chain converges to a unique steady-state vector fixed by that matrix. Both applications turn a dynamic process into a subspace or a fixed-point question.\n",{"module":1111,"moduleNumber":1018,"slug":1112,"lessons":1113},"Eigenvalues and Eigenvectors","eigenvalues",[1114,1119,1124,1129,1134,1139,1144],{"title":1115,"path":1116,"lessonNumber":990,"topics":1117,"summary":1118},"Eigenvectors and Eigenvalues","\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues",[1111],"An eigenvector of a square matrix is a nonzero vector the matrix only stretches; its eigenvalue is the stretch factor. The eigenspace of an eigenvalue is the null space of A minus lambda times the identity, the eigenvalues of a triangular matrix are its diagonal entries, and eigenvectors for distinct eigenvalues are linearly independent.\n",{"title":1120,"path":1121,"lessonNumber":16,"topics":1122,"summary":1123},"The Characteristic Equation","\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation",[1111],"The eigenvalues of a matrix are the roots of its characteristic polynomial det(A minus lambda I). This degree-n polynomial carries an algebraic multiplicity at each repeated root, a nonzero determinant is equivalent to zero not being an eigenvalue, and similar matrices share a characteristic polynomial and hence the same eigenvalues.\n",{"title":1125,"path":1126,"lessonNumber":1006,"topics":1127,"summary":1128},"Diagonalization","\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization",[1111],"A matrix is diagonalizable when it factors as A equals P D P inverse with D diagonal, which happens exactly when it has n linearly independent eigenvectors. The factorization computes matrix powers cheaply, distinct eigenvalues guarantee it, and a repeated eigenvalue permits it only when its eigenspace dimension equals its multiplicity.\n",{"title":1130,"path":1131,"lessonNumber":1012,"topics":1132,"summary":1133},"Eigenvectors and Linear Transformations","\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations",[1111],"Every linear transformation between finite-dimensional spaces has a matrix relative to chosen bases, built from the coordinate vectors of the images of the basis vectors. For a map from a space to itself, an eigenvector basis makes that matrix diagonal, and that change of basis is diagonalization; the matrices similar to A are the representations of the map in every basis.\n",{"title":1135,"path":1136,"lessonNumber":1018,"topics":1137,"summary":1138},"Complex Eigenvalues","\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues",[1111],"A real matrix with no real eigenvalues still has complex ones, occurring in conjugate pairs. A real 2-by-2 matrix with eigenvalue a plus b i is similar to a rotation-scaling matrix, whose rotation angle is the argument of the eigenvalue and whose scale factor is its modulus; the modulus decides whether the trajectories close up, spiral in, or spiral out.\n",{"title":1140,"path":1141,"lessonNumber":1101,"topics":1142,"summary":1143},"Discrete and Continuous Dynamical Systems","\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems",[1111],"Eigenvalues govern the long-term behavior of a system that evolves by x becomes A x or by x prime equals A x. An eigenvector basis decouples both kinds of system into independent scalar equations; the eigenvalues then classify the origin as attractor, repeller, saddle, or spiral, and the dominant eigenpair fixes the growth rate and limiting direction.\n",{"title":1145,"path":1146,"lessonNumber":1107,"topics":1147,"summary":1148},"Iterative Estimates for Eigenvalues","\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method",[1111],"When only a numerical eigenvalue is needed, iteration is preferred over the characteristic polynomial. The power method repeatedly multiplies by A to converge on the dominant eigenvalue and its eigenvector; the Rayleigh quotient sharpens the estimate for symmetric matrices; and the inverse power method targets any eigenvalue near a known guess.\n",{"module":1150,"moduleNumber":1101,"slug":1151,"lessons":1152},"Orthogonality and Least Squares","orthogonality-least-squares",[1153,1158,1163,1168,1173,1178],{"title":1154,"path":1155,"lessonNumber":990,"topics":1156,"summary":1157},"Inner Product, Length, and Orthogonality","\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality",[1150],"The dot product turns the algebra of vectors in R^n into geometry: length, distance, and perpendicularity. The inner product yields the norm, the Pythagorean theorem, and the orthogonal complement, and the null space of a matrix is the orthogonal complement of its row space.\n",{"title":1159,"path":1160,"lessonNumber":16,"topics":1161,"summary":1162},"Orthogonal Sets and Orthogonal Projections","\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections",[1150],"An orthogonal basis makes coordinates trivial: each weight is a single dot product, no linear system required. Orthogonal and orthonormal bases give a direct projection formula onto a line and onto a subspace, the orthogonal decomposition and best-approximation theorems, and the matrix form U U-transpose of a projection.\n",{"title":1164,"path":1165,"lessonNumber":1006,"topics":1166,"summary":1167},"The Gram-Schmidt Process and QR Factorization","\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr",[1150],"Gram-Schmidt turns any basis into an orthogonal one by repeatedly subtracting off projections onto the span already built. Normalizing the result and recording the coefficients factors the matrix as A = QR, with Q orthonormal and R upper triangular, the factorization behind stable least-squares and eigenvalue algorithms.\n",{"title":1169,"path":1170,"lessonNumber":1012,"topics":1171,"summary":1172},"Least-Squares Problems","\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems",[1150],"When Ax = b has no solution, the least-squares solution makes Ax as close to b as possible. The closest Ax is the projection of b onto the column space, and the vector that produces it solves the normal equations A-transpose A x = A-transpose b. Uniqueness, the residual error, and the stabler QR route follow.\n",{"title":1174,"path":1175,"lessonNumber":1018,"topics":1176,"summary":1177},"Applications to Linear Models","\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications",[1150],"Curve fitting is a least-squares problem in statistical notation. The least-squares line, polynomial fits, and multiple regression all reduce to X beta = y with a design matrix X built from the data, solved by the same normal equations.\n",{"title":1179,"path":1180,"lessonNumber":1101,"topics":1181,"summary":1182},"Inner Product Spaces","\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces",[1150],"Promoting the four properties of the dot product to axioms defines an inner product on any vector space, including spaces of functions. Length, distance, orthogonality, Gram-Schmidt, and best approximation all carry over, along with the Cauchy-Schwarz and triangle inequalities and the integral inner product behind Fourier approximation.\n",{"module":1184,"moduleNumber":1107,"slug":1185,"lessons":1186},"Symmetric Matrices, Quadratic Forms, and the SVD","symmetric-quadratic-svd",[1187,1195,1199,1204,1209],{"title":1188,"path":1189,"lessonNumber":990,"topics":1190,"summary":1194},"Diagonalization of Symmetric Matrices","\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices",[1191,1192,1193],"Symmetric Matrices","Quadratic Forms","and the SVD","A symmetric matrix is one that equals its own transpose. Every such matrix can be diagonalized by an orthogonal change of basis, A = PDPᵀ, with real eigenvalues and perpendicular eigenvectors. This is the Spectral Theorem, and it rewrites A as a weighted sum of rank-one projections onto its eigenvectors.\n",{"title":1192,"path":1196,"lessonNumber":16,"topics":1197,"summary":1198},"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms",[1191,1192,1193],"A quadratic form xᵀAx is the second-degree analogue of a linear map, attached to a symmetric matrix A. Orthogonal diagonalization changes variables to the eigenbasis, removing all cross-terms and rotating the form into standard position. The signs of the eigenvalues then classify it as definite or indefinite.\n",{"title":1200,"path":1201,"lessonNumber":1006,"topics":1202,"summary":1203},"Constrained Optimization","\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization",[1191,1192,1193],"Maximizing a quadratic form xᵀAx over the unit sphere has an exact answer: the maximum is the largest eigenvalue of A, attained at its eigenvector, and the minimum is the smallest eigenvalue. Adding orthogonality constraints peels off the eigenvalues in order, characterizing the whole spectrum by optimization.\n",{"title":1205,"path":1206,"lessonNumber":1012,"topics":1207,"summary":1208},"The Singular Value Decomposition","\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition",[1191,1192,1193],"The singular value decomposition factors any m×n matrix as A = UΣVᵀ, with orthogonal U and V and a nonnegative diagonal Σ of singular values. The singular values are the square roots of the eigenvalues of AᵀA, and they describe the matrix geometrically as a rotation, an axiswise stretch, and another rotation, exposing rank, the four fundamental subspaces, and a best low-rank approximation.\n",{"title":1210,"path":1211,"lessonNumber":1018,"topics":1212,"summary":1213},"Applications: Image Processing and Statistics","\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging",[1191,1192,1193],"Principal component analysis diagonalizes the covariance matrix of a data set, producing uncorrelated variables ordered by variance. The leading components capture most of the variation, which reduces dimension, compresses images through low-rank SVD approximation, and connects directly to the singular values of the data matrix.\n",{"module":1215,"moduleNumber":1216,"slug":1217,"lessons":1218},"Numerical Linear Algebra",8,"numerical-linear-algebra",[1219,1224,1229,1234,1239,1244],{"title":1220,"path":1221,"lessonNumber":990,"topics":1222,"summary":1223},"Numerical Thinking and Matrix Computation","\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation",[1215],"Numerical analysis builds efficient discrete algorithms for continuous problems, and its cost is dominated as much by memory traffic as by arithmetic. Block matrix calculus, flop counts, and the BLAS efficiency ratio fix the cost model; triangular and unitary matrices are the two computational building blocks every factorization rests on.\n",{"title":1225,"path":1226,"lessonNumber":16,"topics":1227,"summary":1228},"LU and Cholesky Factorization in Practice","\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky",[1215],"Gaussian elimination, read as a factorization A = LU, turns a linear system into two triangular solves. A single near-zero pivot wrecks it, so partial pivoting reorders rows to pick the largest available pivot and makes the method work for every invertible matrix. For symmetric positive-definite systems, Cholesky halves the cost and needs no pivoting.\n",{"title":1230,"path":1231,"lessonNumber":1006,"topics":1232,"summary":1233},"Conditioning and Floating-Point Arithmetic","\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point",[1215],"A problem's condition number measures how much its answer moves when its data is perturbed, independent of any algorithm. Subtraction is ill-conditioned under cancellation, and for a linear system the amplifier is the matrix condition number κ(A). Floating-point arithmetic supplies the perturbation: every real number is rounded to within a relative machine precision, so even perfect computation inherits an error of order κ times the unit roundoff.\n",{"title":1235,"path":1236,"lessonNumber":1012,"topics":1237,"summary":1238},"Numerical Stability and Backward Error Analysis","\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis",[1215],"An algorithm is backward stable when its computed answer is the exact answer to a slightly perturbed problem. Combined with the condition number this gives the governing rule of thumb: forward error is at most condition times stability. Three cancellation case studies make the point, then the residual-based backward error applies it to Ax = b and shows why partial pivoting keeps Gaussian elimination stable.\n",{"title":1240,"path":1241,"lessonNumber":1018,"topics":1242,"summary":1243},"QR, Householder, and Numerical Least Squares","\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares",[1215],"The least-squares problem reduces to the normal equations, but forming AᵀA squares the condition number and can wreck accuracy. The stable route computes a QR factorization directly on A and solves Rx = Qᵀb. Householder reflectors build that QR one column at a time using length-preserving reflections, the unconditionally backward-stable building block behind every serious least-squares solver.\n",{"title":1245,"path":1246,"lessonNumber":1101,"topics":1247,"summary":1248},"Numerical Eigenvalue Problems and the SVD","\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd",[1215],"Eigenvalues cannot be found by a formula for large matrices, so they are found by iteration. Power and inverse iteration converge to one eigenvector at a rate set by the eigenvalue gap; the QR algorithm sweeps a matrix to Schur form and, with a good shift and a Hessenberg reduction, computes the whole spectrum in cubic time. Singular values follow from the same machinery applied without ever forming AᵀA.\n",{"module":1250,"moduleNumber":1251,"slug":1252,"lessons":1253},"Geometry of Vector Spaces",9,"geometry-of-vector-spaces",[1254,1259,1264,1269,1274],{"title":1255,"path":1256,"lessonNumber":990,"topics":1257,"summary":1258},"Affine Combinations","\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations",[1250],"An affine combination is a linear combination whose weights sum to one. The affine hull of a set is the smallest flat containing it: a point, a line, a plane, or a translated subspace. Homogeneous coordinates turn every affine combination into an ordinary linear combination one dimension up.\n",{"title":1260,"path":1261,"lessonNumber":16,"topics":1262,"summary":1263},"Affine Independence and Barycentric Coordinates","\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates",[1250],"Affine independence is linear independence for the translated or lifted points, and it guarantees each point of an affine hull a unique weight vector. Those weights are barycentric coordinates: centers of mass, ratios of triangle areas, and the interpolation rule behind smooth shading in computer graphics.\n",{"title":1265,"path":1266,"lessonNumber":1006,"topics":1267,"summary":1268},"Convex Combinations and Convex Sets","\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets",[1250],"A convex combination is an affine combination with nonnegative weights, and the convex hull of a set is the smallest convex set containing it. Convex sets are closed under intersection, and Carathéodory's theorem bounds how many points a convex combination in $\\mathbb{R}^n$ ever needs: at most $n+1$.\n",{"title":1270,"path":1271,"lessonNumber":1012,"topics":1272,"summary":1273},"Hyperplanes and Polytopes","\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes",[1250],"A hyperplane is a level set of a linear functional, the set where an inner product equals a constant. Hyperplanes separate disjoint convex sets and support them at their boundaries. Polytopes are convex hulls of finite point sets; their vertices are the extreme points, and a linear functional attains its extremes there.\n",{"title":1275,"path":1276,"lessonNumber":1018,"topics":1277,"summary":1278},"Curves and Surfaces","\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces",[1250],"Bézier curves are affine combinations of control points with polynomial weights, so they lie in the convex hull of those points and bend toward them. The de Casteljau algorithm evaluates them by repeated interpolation, a matrix form factors them for computation, and matching endpoints and tangents joins segments into smooth curves and surfaces.\n",1784263479151]