math.la…

 
Applications of band matrices math.la.a.bandmatrix
Applications to cubic spline math.la.a.cubicspline
Applications to differential equations math.la.a.differentialequation
Applications to error-correcting code math.la.a.errorcorrectingcode
Application Leontief input-output analysis math.la.a.inputoutput
Applications to Markov chains math.la.a.markovchain
Applications to voting and social choice math.la.a.voting
Parametric vector form of the solution set of a system of linear equations math.la.c.linsys.soln_set.vec
Geometric properties of linear transformations math.la.c.lintrans.geometric
Geometric properties of linear transformations on R^2 math.la.c.lintrans.geometric.r2
The condition number of matrix measures how close it is to being singular math.la.c.mat.conditionnumber
A matrix is called ill-conditioned if it is nearly singular math.la.c.mat.illconditioned
Multiplication of block/partitioned matrices math.la.c.mat.mult.block
For matrices, AB=AC does not imply B=C in general. math.la.c.mat.mult.cancellation
Matrix multiplication is not commutative in general. math.la.c.mat.mult.commut
For matrices, AB=0 does not imply A=0 or B=0 in general. math.la.c.mat.mult.zero_divisor
Matrices act as a transformations by multiplying vectors math.la.c.transformation.matrix
Geometric description of span of a set of vectors in R^n (or C^n) math.la.c.vec.span.geometric.rncn
Definition of cross product math.la.d.crossproduct
Definition of distance, coordinate setting math.la.d.distance.coord
Description of the Gram-Schmidt process math.la.d.gramschmidt
Definition of Gram-Schmidt process, arbitrary setting math.la.d.gramschmidt.arb
Definition of inner product, complex entries, coordinate setting math.la.d.innerproduct.complex.coord
Definition of inner product, real entries, coordinate setting math.la.d.innerproduct.real.coord
Definition of the least-squares linear fit to 2-dimensional data math.la.d.leastsquares.line
Definition of linear equation math.la.d.lineqn
Definition of coefficients of a linear equation math.la.d.lineqn.coeff
Definition of solution to a linear equation math.la.d.lineqn.soln
Definition of system of linear equations math.la.d.linsys
Definition of consistent linear system math.la.d.linsys.consistent
Definition of equivalent systems of linear equations math.la.d.linsys.equiv
Definition of homogeneous linear system of equations math.la.d.linsys.homog
Homogeneous linear systems are consistent. math.la.d.linsys.homog.consistent
Definition of nontrivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.nontrivial
Definition of trivial solution to a homogeneous linear system of equations math.la.d.linsys.homog.trivial
Definition of ill-conditioned linear system math.la.d.linsys.ill_conditioned
Definition of inconsistent linear system math.la.d.linsys.inconsistent
Definition of least-squares solution to a linear system math.la.d.linsys.leastsquares
Definition of least-squares error of a linear system math.la.d.linsys.leastsquares.error
Definition of matrix representation of a linear system math.la.d.linsys.mat.repn
Definition of equation operations on a linear system math.la.d.linsys.op
Definition of solution to a system of linear equations math.la.d.linsys.soln
Definition of solution set of a system of linear equations math.la.d.linsys.soln_set
Definition of basic/dependent/leading variable in a linear system math.la.d.linsys.variable.dependent
Definition of free/independent variable in a linear system math.la.d.linsys.variable.independent
Definition of linear transformation, arbitrary vector space math.la.d.lintrans.arb
Definition of codomain of a linear transformation, coordinate vector space math.la.d.lintrans.codomain.coord
Definition of linear transformation, coordinate vector space math.la.d.lintrans.coord
Definition of domain of a linear transformation, coordinate vector space math.la.d.lintrans.domain.coord
Definition of eigenvalue of a linear transformation, arbitrary vector space math.la.d.lintrans.eig.arb
Definition of eigenvalue of a linear transformation, coordinate vector space math.la.d.lintrans.eig.coord
Definition of image (of a point) under a linear transformation, coordinate vector space math.la.d.lintrans.image.coord
Definition of one-to-one/injective linear transformation, arbitrary vector space math.la.d.lintrans.injective.arb
Definition of one-to-one/injective linear transformation, coordinate vector space math.la.d.lintrans.injective.coord
Definition of invariant subspace under linear transformation math.la.d.lintrans.invariant_subspace
Definition of invertible linear transformation, arbitrary vector space math.la.d.lintrans.invertible.arb
Definition of invertible linear transformation, coordinate vector space math.la.d.lintrans.invertible.coord
Definition of kernel of linear transformation, arbitrary vector space math.la.d.lintrans.kernel.arb
Definition of kernel of linear transformation, coordinate vector space math.la.d.lintrans.kernel.coord
Definition of the standard matrix for a linear transformation, coordinate setting math.la.d.lintrans.mat.basis.standard.coord
Definition of matrix representation of a linear transformation with respect to bases of the spaces, arbitrary vector space math.la.d.lintrans.mat.repn.arb
Definition of matrix representation of a linear transformation, coordinate vector space math.la.d.lintrans.mat.repn.coord
Definition of matrix representation of a linear transformation from a vector space to itself, with respect to basis of the space, arbitrary vector space math.la.d.lintrans.mat.repn.self.arb
Definition of matrix representation of a composition of linear transformations, arbitrary vector space math.la.d.lintrans.mat_repn.composition.arb
Definition of nilpotent linear transformation math.la.d.lintrans.nilpotent
Definition of nullity of a linear transformation math.la.d.lintrans.nullity
Definition of pre-image of linear transformation, arbitrary vector space math.la.d.lintrans.preimage.arb
Definition of pre-image (of a point) under a linear transformation, coordinate vector space math.la.d.lintrans.preimage.coord
Definition of range of linear transformation, arbitrary vector space math.la.d.lintrans.range.arb
Definition of range of linear transformation, coordinate vector space math.la.d.lintrans.range.coord
Definition of rank of a linear transformation math.la.d.lintrans.rank
Definition of onto/surjective linear transformation, arbitrary vector space math.la.d.lintrans.surjective.arb
Definition of onto/surjective linear transformation, coordinate vector space math.la.d.lintrans.surjective.coord
Definition of matrix math.la.d.mat
Definition of augmented matrix of a linear system math.la.d.mat.augmented
Definition of band matrix math.la.d.mat.band
Definition of block/partitioned matrix math.la.d.mat.block
Definition of block diagonal matrix math.la.d.mat.block_diagonal
Definition of characteristic polynomial of a matrix math.la.d.mat.charpoly
Definition of characteristic equation of a matrix math.la.d.mat.charpoly.eqn
Definition of Cholesky decomposition math.la.d.mat.cholesky
Definition of adjugate/classical adjoint of a matrix math.la.d.mat.classicaladjoint
Definition of coefficient matrix of a linear system math.la.d.mat.coeff
Definition of cofactor of a matrix math.la.d.mat.cofactor
Definition of column space of a matrix math.la.d.mat.col_space
Definition of determinant of a matrix as a cofactor expansion across the first row math.la.d.mat.det.cofactor
Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form. math.la.d.mat.det.echelon
Definition of diagonal matrix math.la.d.mat.diagonal
Definition of diagonalizable matrix math.la.d.mat.diagonalizable
Definition of orthogonally diagonalizable matrix math.la.d.mat.diagonalizable.orthogonally
Definition of matrix diagonalization math.la.d.mat.diagonalization
Definition of (echelon matrix/matrix in (row) echelon form) math.la.d.mat.echelon
Definition of (row) echelon form of a matrix math.la.d.mat.echelon.of
Definition of eigenvalue(s) of a matrix math.la.d.mat.eig
Definition of multiplicity of an eigenvalue math.la.d.mat.eig.multiplicity
Definition of eigenspace(s) of a matrix math.la.d.mat.eigsp
Definition of eigenvector(s) of a matrix math.la.d.mat.eigvec
Definition of elementary matrix math.la.d.mat.elementary
Elementary matrices are invertible. math.la.d.mat.elementary.inv
Notation for entry of matrix math.la.d.mat.entry
Definition of matrix equation math.la.d.mat.eqn
Definition of equality of matrices math.la.d.mat.equal
Definition of Hermitian matrix math.la.d.mat.hermitian
Definition of Hessenberg form math.la.d.mat.hessenberg
Definition of identity matrix math.la.d.mat.identity
Definition of matrix inverse math.la.d.mat.inv
Definition of generalized inverse of a matrix math.la.d.mat.inv.generalized
Definition of Jordan form math.la.d.mat.jordan
Definition of LU decomposition math.la.d.mat.lu
Definition of reduced LU decomposition math.la.d.mat.lu.reduced
Definition of m by n matrix math.la.d.mat.m_by_n
Definition of minimal polynomial of a matrix math.la.d.mat.minpoly
Definition of matrix multiplication in terms of column vectors math.la.d.mat.mult.col
Definition of matrix multiplication, each entry separately math.la.d.mat.mult.coord
Row operations are given by multiplication by elementary matrices. math.la.d.mat.mult.elementary
Definition of nonsingular matrix: matrix is invertible math.la.d.mat.nonsingular.inv
Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution math.la.d.mat.nonsingular.z
Definition of matrix null space (left) math.la.d.mat.null_space.left
Definition of matrix null space (right) math.la.d.mat.null_space.right
Definition of nullity of a matrix math.la.d.mat.nullity
Definition of orthogonal matrix math.la.d.mat.orthogonal
Definition of pivot math.la.d.mat.pivot
Definition of pivot column math.la.d.mat.pivot_col
Definition of pivot position math.la.d.mat.pivot_position
Definition of positive-definite matrix math.la.d.mat.positive_definite
Definition of QR decomposition math.la.d.mat.qr
Definition of rank of a matrix math.la.d.mat.rank
Definition of rank factorization of a matrix math.la.d.mat.rank_factorization
Definition of rational form math.la.d.mat.rational
Definition of leading entry in a row of a matrix math.la.d.mat.row.leading
Definition of row equivalent matrices math.la.d.mat.row_equiv
Definition of row operations on a matrix math.la.d.mat.row_op
Definition of row reduce a matrix math.la.d.mat.row_reduce
Definition of row space of a matrix math.la.d.mat.row_space
Row equivalent matrices have the same row space. math.la.d.mat.row_space.row_equiv
Definition of matrix in reduced row echelon form math.la.d.mat.rref
Definition of reduced row echelon form of a matrix math.la.d.mat.rref.of
Definition of matrix-scalar multiplication math.la.d.mat.scalar.mult
Definition of Schur triangulation math.la.d.mat.schur
Definition of similar matrices math.la.d.mat.similar
Definition of similarity transform math.la.d.mat.similar.transform
Definition of singular matrix math.la.d.mat.singular
Definition of size of a matrix math.la.d.mat.size
Definition of square matrix math.la.d.mat.square
Definition of sum of matrices math.la.d.mat.sum
Definition of singular value decomposition (SVD) math.la.d.mat.svd
Definition of symmetric matrix math.la.d.mat.symmetric
Definition of the diagonal of a matrix math.la.d.mat.thediagonal
Definition of transpose of a matrix math.la.d.mat.transpose
Definition of a lower triangular matrix math.la.d.mat.triangular.lower
Definition of an upper triangular matrix math.la.d.mat.triangular.upper
Definition of unitary matrix math.la.d.mat.unitary
Definition of Vandermonde matrix math.la.d.mat.vandermonde
Definition of matrix-vector product, as a linear combination of column vectors math.la.d.mat.vec.prod
Definition of matrix-vector product, each entry separately math.la.d.mat.vec.prod.coord
Definition of zero matrix math.la.d.mat.zero
Definition of (orthogonal) projection onto a subspace math.la.d.projection.subspace
Definition of quadratic form, orthonormal diagonalization, principal axes math.la.d.quadraticform
Definition of scalar, coordinate vector space math.la.d.scalar
Definition of orthogonal basis of a (sub)space math.la.d.subspace.basis.orthogonal
Definition of orthonormal basis of a (sub)space math.la.d.subspace.basis.orthonormal
Definition of orthogonal complement of a subspace math.la.d.subspace.orthogonal_complement
Definition of bijective linear transformation math.la.d.trans.bijective
Definition of injective/one-to-one linear transformation math.la.d.trans.injective
Definition of surjective/onto linear transformation math.la.d.trans.surjective
Definition of vector addition, arbitrary vector space math.la.d.vec.add.arb
Definition of vector, arbitrary vector space math.la.d.vec.arb
Definition of column vector, coordinate vector space math.la.d.vec.col.coord
Definition of entry/component of a vector, coordinate vector space math.la.d.vec.component.coord
Definition of constant vector of a linear system math.la.d.vec.constant
Definition of vector, coordinate vector space math.la.d.vec.coord
Definition of orthogonal vectors, arbitrary setting math.la.d.vec.distance.arb
Definition of equality of vectors, coordinate vector space math.la.d.vec.equal.coord
Definition of the imaginary part of a vector in C^n math.la.d.vec.imaginary.cn
Definition of inner product, arbitrary setting math.la.d.vec.innerproduct.arb
Definition of linear combination of vectors, arbitrary vector space math.la.d.vec.lincomb.arb
Definition of linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.coord
Definition of weights in a linear combination of vectors, coordinate vector space math.la.d.vec.lincomb.weight.coord
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors, arbitrary vector space. math.la.d.vec.lindep.arb
Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.d.vec.lindep.coord
Definition of linear dependence relation math.la.d.vec.lindep.relation
Definition of linearly indepentent set of vectors: if a linear combination is zero, then every coefficient is zero, arbitrary vector space. math.la.d.vec.linindep.arb
Definition of linearly independent set of vectors: if a linear combination is zero, then every coefficient is zero, coordinate vector space. math.la.d.vec.linindep.coord
Definition of length/norm of a vector, arbitrary setting math.la.d.vec.norm.arb
Definition of norm/length of a vector, coordinate setting math.la.d.vec.norm.coord
Definition of two vectors being orthogonal math.la.d.vec.orthogonal
Definition of orthogonal vectors, arbitrary setting math.la.d.vec.orthogonal.arb
Definition of orthogonal set of vectors math.la.d.vec.orthogonal_set
Definition of orthonormal set of vectors math.la.d.vec.orthonormal_set
Definition of parallel vectors, arbitrary setting math.la.d.vec.parallel.arb
Definition of (orthogonal) projection of one vector onto another vector math.la.d.vec.projection
Definition of orthogonal projection onto a subspace math.la.d.vec.projection.subspace
Definition of the real part of a vector in C^n math.la.d.vec.real.cn
Definition of vector-scalar multiplication, arbitrary vector space math.la.d.vec.scalar.mult.arb
Definition of vector-scalar multiplication, coordinate vector space math.la.d.vec.scalar.mult.coord
Definition of size of a vector, coordinate vector space math.la.d.vec.size.coord
Definition of solution vector of a linear system math.la.d.vec.solution
Definition of span of a set of vectors, arbitrary vector space math.la.d.vec.span.arb
Definition of span of a set of vectors, coordinate vector space math.la.d.vec.span.coord
Definition of subspace spaned by a set of vectors, arbitrary vector space math.la.d.vec.span.subspace.arb
Definition of subspace spanned by a set of a set of vectors, coordinate vector space math.la.d.vec.span.subspace.coord
Definition of a vector being orthogonal to a subspace math.la.d.vec.subspace.orthogonal
Definition of vector sum/addition, coordinate vector space math.la.d.vec.sum.coord
Definition of unit vector, coordinate setting math.la.d.vec.unit.coord
Definition of zero vector, coordinate vector space math.la.d.vec.z.coord
Axioms of a vector space, arbitrary vector space math.la.d.vsp.axioms.arb
Axioms of a vector space, coordinate vector space math.la.d.vsp.axioms.coord
Definition of basis of a vector space (or subspace), arbitrary vector space math.la.d.vsp.basis.arb
Definition of basis of a vector space (or subspace), coordinate vector space math.la.d.vsp.basis.coord
Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n) math.la.d.vsp.basis.coord.change.rncn
Definition of coordinate vector/mapping relative to a given basis, arbitrary vector space math.la.d.vsp.basis.coord.vector.arb
Definition of coordinates relative to a given basis, arbitrary vector space math.la.d.vsp.basis.relative.arb
Definition of coordinates relative to a given basis, coordinate vector space math.la.d.vsp.basis.relative.coord
Definition of the standard basis of the polynomials of degree at most n math.la.d.vsp.basis.standard.leq_n
Definition of the standard basis of R^n (or C^n) math.la.d.vsp.basis.standard.rncn
Definition of change of corrdinates matrix between two bases, arbitrary vector space math.la.d.vsp.change_of_basis.arb
Definition of dimension of a vector space (or subspace), arbitrary vector space math.la.d.vsp.dim.arb
Definition of dimension of a vector space (or subspace), coordinate vector space math.la.d.vsp.dim.coord
Definition of dimension of a vector space (or subspace) being finite or infinite, arbitrary vector space math.la.d.vsp.dim.finite_infinite.arb
Definition of inner product space, arbitrary setting math.la.d.vsp.innerproduct.arb
Definition of isomorphism between vector spaces math.la.d.vsp.isomorphism
Definition of orthogonal subspaces, arbitrary setting math.la.d.vsp.orthogonal.arb
Definition of spanning set for a subspace, arbitrary vector space math.la.d.vsp.span.set.arb
Definition of subspace, arbitrary vector space math.la.d.vsp.subspace.arb
Definition of subspace, coordinate vector space math.la.d.vsp.subspace.coord
Definition of intersection of subspaces, arbitrary vector space math.la.d.vsp.subspace.intersection.arb
Definition of sum of subspaces, arbitrary vector space math.la.d.vsp.subspace.sum.arb
Definition of zero subspace, arbitrary vector space math.la.d.vsp.subspace.z
Definition of the zero/trivial subspace math.la.d.vsp.subspace.z.coord
The additive inverse of a vector is called the negative of the vector. math.la.d.vsp.vector.negative
Example of using the echelon form to determine if a linear system is consistent. math.la.e.echelon.consistent
Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.homog.row_reduce.i
Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix, in the case of one solution math.la.e.linsys.3x3.soln.homog.row_reduce.o
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of infinitely many solutions math.la.e.linsys.3x3.soln.row_reduce.i
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of one solution math.la.e.linsys.3x3.soln.row_reduce.o
Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix, in the case of no solutions math.la.e.linsys.3x3.soln.row_reduce.z
Non-example of a linear transformation math.la.e.lintrans.not
Example of a linear transformation on R^2: projection math.la.e.lintrans.projection.r2
Example of a linear transformation on R^2: rotation math.la.e.lintrans.rotation.r2
Example of a linear transformation on R^3: rotation math.la.e.lintrans.rotation.r3
Example of a linear transformation on R^2: shear math.la.e.lintrans.shear.r2
Example of (echelon matrix/matrix in (row) echelon form) math.la.e.mat.echelon
Example of putting a matrix in echelon form math.la.e.mat.echelon.of
Example of putting a matrix in echelon form and identifying the pivot columns math.la.e.mat.echelon.of.pivot
Example of solving a 3-by-3 homogeneous matrix equation math.la.e.mat.eqn.3x3.homog.solve
Example of solving a 3-by-3 matrix equation math.la.e.mat.eqn.3x3.solve
Example of multiplying matrices math.la.e.mat.mult
Example of multiplying 2x2 matrices math.la.e.mat.mult.2x2
Example of multiplying 3x3 matrices math.la.e.mat.mult.3x3
Example of multiplying nonsquare matrices math.la.e.mat.mult.nonsquare
Example of matrix-vector product, as a linear combination of column vectors math.la.e.mat.vec.prod
Example of matrix-vector product, each entry separately math.la.e.mat.vec.prod.coord
Example of linear combination of vectors in R^2 math.la.e.vec.lincomb.r2
Example of writing a given vector in R^3 as a linear combination of given vectors math.la.e.vec.lincomb.weight.solve.r3
Determine if a particular set of vectors in R^3 in linearly independent math.la.e.vec.linindep.r3
Example of vector-scalar multiplication in R^2 math.la.e.vec.scalar.mult.r2
Determine if a particular vector is in the span of a set of vectors math.la.e.vec.span.of
Determine if a particular vector is in the span of a set of vectors in R^3 math.la.e.vec.span.of.r3
Determine if a particular set of vectors spans R^3 math.la.e.vec.span.r3
Example of a sum of vectors interpreted geometrically in R^2 math.la.e.vec.sum.geometric.r2
Cramer's rule math.la.t.cramer
The echelon form can be used to determine if a linear system is consistent. math.la.t.echelon.consistent
The inverse of a matrix can be used to solve a linear system. math.la.t.eqn.mat.inv
Equivalence theorem: the columns of A are a basis for R^n (or C^n). math.la.t.equiv.col.basis
Equivalence theorem: the dimension of the column space of A is n. math.la.t.equiv.col.dim
Equivalence theorem: the columns of A are linearly independent. math.la.t.equiv.col.linindep
Equivalence theorem: the columns of A span R^n (or C^n). math.la.t.equiv.col.span
Equivalence theorem: the determinant of A is nonzero. math.la.t.equiv.det
Equivalence theorem: the matrix A does not have zero as an eigenvalue. math.la.t.equiv.eig
Equivalence theorem: the matrix A row-reduces to the identity matrix. math.la.t.equiv.identity
Equivalence theorem: the matrix A has an inverse. math.la.t.equiv.inv
Equivalence theorem: the matrix A has a left inverse. math.la.t.equiv.inv.left
Equivalence theorem: the matrix A has a right inverse. math.la.t.equiv.inv.right
Equivalence theorem: the linear transformation given by T(x)=Ax is one-to-one/injective. math.la.t.equiv.lintrans.injective
Equivalence theorem: the linear transformation given by T(x)=Ax has an inverse. math.la.t.equiv.lintrans.inv
Equivalence theorem: the linear transformation given by T(x)=Ax is onto/surjective. math.la.t.equiv.lintrans.surjective
Equivalence theorem: the equation Ax=b has a solution for all b. math.la.t.equiv.mat.eqn
Equivalence theorem: the equation Ax=0 has only the trivial solution. math.la.t.equiv.mat.eqn.homog
Equivalence theorem: the equation Ax=b has a solution for all b. math.la.t.equiv.mat.eqn.unique
Equivalence theorem: the nullity of the matrix A is zero. math.la.t.equiv.nullity
Equivalence theorem: the null space of the matrix A is {0}. math.la.t.equiv.nullspace
Equivalence theorem: the matrix A has rank n. math.la.t.equiv.rank
Equivalence theorem: the rows of A are a basis for R^n (or C^n). math.la.t.equiv.row.basis
Equivalence theorem: the rows of A are linearly independent. math.la.t.equiv.row.linindep
Equivalence theorem: there is a pivot position in every row of A. math.la.t.equiv.row.pivot
Equivalence theorem: the rows of A span R^n (or C^n). math.la.t.equiv.row.span
Equivalence theorem: the transpose of the matrix A has an inverse. math.la.t.equiv.transpose.inv
The Gram-Schmidt process converts a basis into an orthogonal basis. math.la.t.gramschmidt
The standard inner product is commutative, coordinate setting. math.la.t.innerproduct.commutative.coord
The standard inner product commutes with real scalar multiplication, coordinate setting. math.la.t.innerproduct.commutative.scalar.real.coord
The standard inner product distributes over addition, coordinate setting. math.la.t.innerproduct.distributive.coord
The standard inner product of a vector with itself is non-negative, coordinate setting. math.la.t.innerproduct.self.positive.coord
The standard inner product of a vector with itself is zero only for the zero vector, coordinate setting. math.la.t.innerproduct.self.z.coord
Formula for the least-squares linear fit to 2-dimensional data math.la.t.leastsquares.line
A consistent system with more variables than equations has infinitely many solutions. math.la.t.linsys.consistent.i
A homog system with more variables than equations has infinitely many solutions. math.la.t.linsys.homog.i
A homogeneous system has a nontrivial solution if and only if it has a free variable. math.la.t.linsys.homog.nontrivial
The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation. math.la.t.linsys.homog.soln_preimage
Formula for computing the least squares solution to a linear system. math.la.t.linsys.leastsquares
Formula for computing the least squares solution to a linear system, in terms of the QR factorization of the coefficient matrix. math.la.t.linsys.leastsquares.qr
The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent. math.la.t.linsys.leastsquares.unique
The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system. math.la.t.linsys.nonhomog.particular_plus_homog
Equation operations on a linear system give an equivalent system. math.la.t.linsys.op
A linear system is equivalent to a vector equation. math.la.t.linsys.vec
Linear systems have zero, one, or infinitely many solutions. math.la.t.linsys.zoi
The determinant of the matrix of a linear transformation is the factor by which the area/volume changes. math.la.t.lintrans.det.volume
A linear transformation is one-to-one/injective if and only if only 0 is mapped to 0. math.la.t.lintrans.injective.ker
A linear transformation is one-to-one/injective if and only if the columns of its matrix are linearly independent. math.la.t.lintrans.injective.linindep
A linear transformation of a linear combination is the linear combination of the linear transformation math.la.t.lintrans.lincomb
A linear transformation is given by a matrix with respect to a given basis. math.la.t.lintrans.mat.basis
A linear transformation is given by a matrix whose columns are the images of the standard basis vectors, coordinate setting. math.la.t.lintrans.mat.basis.standard.coord
Matrix representation of a composition of linear transformations is given by a matrix product, coordinate vector space math.la.t.lintrans.mat_repn.composition.coord
A linear transformation is onto/surjective if and only if the columns of its matrix span the codomain. math.la.t.lintrans.surjective.col_span
A linear transformation is onto/surjective if and only if the columns of its matrix span the codomain. math.la.t.lintrans.surjective.span
A linear transformation maps zero to zero. math.la.t.lintrans.z
Matrix addition is commutative and associative. math.la.t.mat.add.commut_assoc
The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation. math.la.t.mat.charpoly.eig
A matrix with real entries and orthonormal columns preserves dot products. math.la.t.mat.col.orthonormal.dot.rn
A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose. math.la.t.mat.col.orthonormal.inv.rn
A matrix with real entries and orthonormal columns preserves norms. math.la.t.mat.col.orthonormal.norm.rn
The pivot columns of a matrix are a basis for the column space. math.la.t.mat.col_space.pivot
The column space of an m-by-n matrix is a subspace of R^m (or C^m) math.la.t.mat.col_space.rncn
Formula for the determinant of a 2-by-2 matrix. math.la.t.mat.det.2x2
Formula for the determinant of a 3-by-3 matrix. math.la.t.mat.det.3x3
The determinant of a matrix can be computed as a cofactor expansion across any row or down any column. math.la.t.mat.det.cofactor
The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns. math.la.t.mat.det.col.volume
The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form. math.la.t.mat.det.echelon
Theorem describing the effect of elementary row operations on the determinant of a matrix. math.la.t.mat.det.elementaryoperations
If A and B are n-by-n matrices, then det(AB)=det(A)det(B). math.la.t.mat.det.product
A matrix and its transpose have the same determinant. math.la.t.mat.det.transpose
The determinant of a triangular matrix is the product of the entries on the diagonal. math.la.t.mat.det.trianglar
An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors. math.la.t.mat.diagonalizable
An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n). math.la.t.mat.diagonalizable.basis
An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely, and the dimension of each eigenspace equals the multiplicity of the eigenvalue. math.la.t.mat.diagonalizable.charpoly
An n-by-n matrix with n distinct eigenvalues is diagonalizable. math.la.t.mat.diagonalizable.distinct
An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n. math.la.t.mat.diagonalizable.eigenspace
A matrix is orthogonally diagonalizable if and only if it is symmetric. math.la.t.mat.diagonalizable.orthogonally
A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns. math.la.t.mat.diagonalized_by
The dimension of a eigenspace is less than or equal to the multiplicity of the eigenvalue. math.la.t.mat.eig.multiplicity.eigenspace
The eigenvalues of a triangular matrix are the entries on the main diagonal. math.la.t.mat.eig.triangular
Eigenvectors with distinct eigenvalues are linearly independent. math.la.t.mat.eigvec.linindep
Theorem describing the determinants of elementary matrices. math.la.t.mat.elementary.det
The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A. math.la.t.mat.eqn.lincomb
A matrix equation is equivalent to a linear system math.la.t.mat.eqn.linsys
Formula for the inverse of a 2-by-2 matrix. math.la.t.mat.inv.2x2
The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix. math.la.t.mat.inv.augmented
The inverse of a matrix can be expressed in terms of its matrix of cofactors. math.la.t.mat.inv.cofactors
Matrix inverse is an involution. math.la.t.mat.inv.involution
For n-by-n invertible matrices A and B, the product AB is invertible, and (AB)^-1=B^-1 A^-1. math.la.t.mat.inv.shoesandsocks
Matrix transpose commutes with matrix inverse. math.la.t.mat.inv.transpose
Matrix inverses are unique: if A and B are square matrices, then AB=I implies that A=B^-1 and B=A^-1. math.la.t.mat.inv.unique
Algorithm for computing an LU decomposition math.la.t.mat.lu
Matrix multiplication is associative. math.la.t.mat.mult.assoc
Matrix multiplication is distributive over matrix addition. math.la.t.mat.mult.distributive
The identity matrix is the identity for matrix multiplication. math.la.t.mat.mult.identity
Matrix multiplication can be viewed as the dot product of a row vector of column vectors with a column vector of row vectors math.la.t.mat.mult.row.col
The transpose of a product of matrices is the product of the transposes in reverse order. math.la.t.mat.mult.transpose
The null space of a matrix is a subspace of R^n (or C^n). math.la.t.mat.null_space.rncn
The QR decomposition of a nonsingular matrix exists. math.la.t.mat.qr
If A is a matrix, then the rank of A plus the nullity of A equals the number of columns of A. math.la.t.mat.ranknullity
Formula for diagonalizing a real 2-by-2 matrix with a complex eigenvalue. math.la.t.mat.real.diagonalize.complex.2x2
A matrix with real entries has eigenvalues occurring in conjugate pairs. math.la.t.mat.real.eig.cn
Matrix describing a rotation of the plane math.la.t.mat.rotation
The null space of a matrix is the orthogonal complement of the column space. math.la.t.mat.row.null.orthogonal_complement
Row equivalent matrices represent equivalent linear systems math.la.t.mat.row_equiv.linsys
The row space and the column space of a matrix have the same dimension. math.la.t.mat.row_space.col_space
Every matrix is row-equivalent to a matrix in reduced row echelon form. math.la.t.mat.rref.exists
Every matrix is row-equivalent to only one matrix in reduced row echelon form. math.la.t.mat.rref.unique
Matrix-scalar multiplication is commutative, associative, and distributive. math.la.t.mat.scalar.mult.commut_assoc
Matrix-scalar product is commutative math.la.t.mat.scalar.prod.commut
Similar matrices have the same eigenvalues and the same characteristic polynomials. math.la.t.mat.similar.eig
Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal. math.la.t.mat.symmetric.eig.orthogonal
The spectral theorem for symmetric matrices math.la.t.mat.symmetric.spectral
Formula for the spectral decomposition for a symmetric matrix math.la.t.mat.symmetric.spectraldecomposition
Matrix transpose is an involution. math.la.t.mat.transpose.involution
Matrix-vector multiplication is a linear transformation. math.la.t.mat.vec.mult.lintrans
Matrix-vector product is associative math.la.t.mat.vec.prod.assoc
The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set. math.la.t.rref.pivot.free
The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions. math.la.t.rref.pivot.oi
Formula for the coordinates of a vector with respect to an orthogonal basis. math.la.t.subspace.basis.orthogonal
Formula for the coordinates of the projection of a vector onto a subspace, with respect to an orthonormal basis. math.la.t.subspace.basis.orthonormal
The orthogonal complement of a subspace is a subspace. math.la.t.subspace.orthogonal_complement
A vector is in the orthogonal complement of a subspace if and only if it is orthogonal to every vector in a basis of the subspace. math.la.t.subspace.orthogonal_complement.basis
The vector space properties of R^n (or C^n) math.la.t.vec.axioms.rncn
The Cauchy-Schwartz inequality, arbitrary setting math.la.t.vec.cauchyschwartz.arb
The Cauchy-Schwarz inequality, arbitrary setting math.la.t.vec.cauchyschwarz.arb
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, arbitrary vector space. math.la.t.vec.lindep.arb
Theorem: a set of vectors is linearly dependent if and only if one of the vectors can be written as a linear combination of the other vectors, coordinate vector space. math.la.t.vec.lindep.coord
If a set of vectors in R^n (or C^n) contains more than n elements, then the set is linearly dependent. math.la.t.vec.lindep.more.rncn
A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other. math.la.t.vec.lindep.two
If a set of vectors contains the zero vector, then the set is linearly dependent. math.la.t.vec.lindep.zero
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is zero, then every coefficient is zero, arbitrary vector space. math.la.t.vec.linindep.arb
Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is zero, then every coefficient is zero, coordinate vector space. math.la.t.vec.linindep.coord
Two vectors are orthogonal if and only if the Pythagorean Theorem holds. math.la.t.vec.orthogonal
An orthogonal set of nonzero vectors is linearly independent. math.la.t.vec.orthogonal_set.linindep
Formula for the (orthogonal) projection of one vector onto another vector math.la.t.vec.projection
The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector. math.la.t.vec.projection.closest
The projection of a vector which is in a subspace is the vector itself. math.la.t.vec.projection.element
A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace. math.la.t.vec.projection.subspace
Vector sum/addition interpreted geometrically in R^n (or C^n) math.la.t.vec.sum.geometric.rncn
The triangle inequality, arbitrary setting math.la.t.vec.triangle.arb
Each vector can be written uniquely as a linear combination of vectors from a given basis. math.la.t.vsp.basis.coord.unique
The coordinate vector/mapping relative to a given basis is a bijective linear mapping to R^n (or C^n). math.la.t.vsp.basis.coord.vector.arb
The set of all polynomials of degree at most n is a vector space. math.la.t.vsp.change_of_basis.arb
Every basis for a vector space contains the same number of elements, arbitrary vector space. math.la.t.vsp.dim.arb
If a vector space has dimension n, then any subset of n vectors that is linearly independent must be a basis, arbitrary vector space. math.la.t.vsp.dim.linindep.arb
If a vector space has dimension n, then any subset of n vectors that is linearly independent must be a basis, coordinate vector space. math.la.t.vsp.dim.linindep.coord
A set of vectors containing more elements than the dimension of the space must be linearly dependent, arbitrary vector space. math.la.t.vsp.dim.more.lindep.arb
If a vector space has dimension n, then any subset set of n vectors that spans the space must be a basis, arbitrary vector space. math.la.t.vsp.dim.span.arb
If a vector space has dimension n, then any subset set of n vectors that spans the space must be a basis, coordinate vector space. math.la.t.vsp.dim.span.coord
Any linearly independent set can be expanded to a basis for the (sub)space, arbitrary vector space. math.la.t.vsp.linindep.basis.arb
The zero scalar multiplied by any vector equals the zero vector. math.la.t.vsp.scalar.mult.z
A set of nonzero vectors contains (as a subset) a basis for its span. math.la.t.vsp.span.basis
Removing a linearly dependent vector from a set does not change the span of the set. math.la.t.vsp.span.lindep
The dimension of a subspace is less than or equal to the dimension of the whole space, arbitrary vector space. math.la.t.vsp.subspace.dim.arb
The intersection of subspaces is a subspace, arbitrary vector space. math.la.t.vsp.subspace.intersection.arb
The sum of subspaces is a subspace, arbitrary vector space. math.la.t.vsp.subspace.sum.arb
The zero vector multiplied by any scalar equals the zero vector. math.la.t.vsp.vector.mult.z
The negative of a vector equals the vector multiplied by -1. math.la.t.vsp.vector.negative