- Abstract vector spaces
- A consistent system with more variables than equations has infinitely many solutions.
- Adding a multiple of one row to another row does not change the determinant.
- Addition
- A determinant is a multilinear function
- A diagonalizable matrix is diagonalized by a matrix having the eigenvectors as columns.
- Adjoining an element not in the span of a linearly independent set gives another linearly independent set.
- A homogeneous system has a nontrivial solution if and only if it has a free variable.
- A homogeneous system with more variables than equations has infinitely many solutions.
- Algebraic properties of R^n (or C^n)
- Algorithm for computing an LU decomposition
- A linear system is equivalent to a matrix equation.
- A linear system is equivalent to a vector equation.
- A linear transformation has a representation as an upper triangular matrix.
- A linear transformation has the same eigenvalues and eigenvectors as any matrix representation.
- A linear transformation is determined by its action on a basis.
- A linear transformation is diagonalizable if there is a basis such that each element is an eigenvector of the transformation.
- A linear transformation is given by a matrix whose columns are the images of the standard basis vectors
- A linear transformation is given by a matrix with respect to a given basis.
- A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces
- A linear transformation is injective on its generalized range space.
- A linear transformation is invertible if and only if it is injective and surjective
- A linear transformation is onto if and only if its rank equals the number of rows in any matrix representation.
- A linear transformation is surjective if and only if the columns of its matrix span the codomain.
- A linear transformation is surjective if and only if the image of a basis is a spanning set
- A linear transformation is surjective if and only if the rank equals the dimension of the codomain.
- A linear transformation maps 0 to 0.
- A linear transformation of a linear combination is the linear combination of the linear transformation
- A linear transformation on a finite dimentional nontrivial vector space has at least one eigenvalue.
- All echelon forms of a linear system have the same free variables
- A matrix and its transpose have the same determinant.
- A matrix and its transpose have the same eigenvalues/characteristic polynomial.
- A matrix A with real entries has orthonormal columns if and only if A inverse equals A transpose.
- A matrix equation is equivalent to a linear system
- A matrix is called ill-conditioned if it is nearly singular
- A matrix is nilpotent if and only if its only eigenvalue is 0.
- A matrix is orthogonally diagonalizable if and only if it is normal (The principal axis theorem).
- A matrix is orthogonally diagonalizable if and only if it is symmetric.
- A matrix of rank k is equivalent to a matrix with 1 in the first k diagonal entries and 0 elsewhere.
- A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.
- A matrix with a 0 row/column has determinant 0
- A matrix with real entries and orthonormal columns preserves dot products.
- A matrix with real entries and orthonormal columns preserves norms.
- A matrix with real entries has eigenvalues occurring in conjugate pairs.
- A matrix with two equal rows/columns has determinant 0
- An eigenspace of a matrix is a nontrivial subspace.
- An eigenspace of a matrix is the null space of a related matrix.
- An n-by-n matrix is diagonalizable if and only if it has n linearly independent eigenvectors.
- An n-by-n matrix is diagonalizable if and only if the characteristic polynomial factors completely
- An n-by-n matrix is diagonalizable if and only if the sum of the dimensions of the eigenspaces equals n.
- 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).
- An n-by-n matrix nas n (complex) eigenvalues
- An n-by-n matrix with n distinct eigenvalues is diagonalizable.
- A nonempty subset of a vector space is a subspace if and only if it is closed under linear combinations
- A nonsingular matrix can be written as a product of elementary matrices.
- An orthogonal set of nonzero vectors is linearly independent.
- Any linearly independent set can be expanded to a basis for the (sub)space
- Any matrix times the 0 matrix equals the 0 matrix.
- Any vector space is the direct sum of the generalized kernel and gneralized range of a linear transformation on that space.
- Application Leontief input-output analysis
- Applications
- Applications of band matrices
- Applications to cubic spline
- Applications to differential equations
- Applications to error-correcting code
- Applications to Markov chains
- Applications to voting and social choice
- A scalar multiple of a linear transformation is a linear transformation
- A set is a basis if each vector can be written uniquely as a linear combination.
- A set is linearly independent if and only if the set of coordinate vectors with respect to any basis is linearly independent.
- A set of nonzero vectors contains (as a subset) a basis for its span.
- A set of two vectors is linearly dependent if and only if neither is a scalar multiple of the other.
- A set of vectors containing fewer elements than the dimension of the space cannot span
- A set of vectors containing more elements than the dimension of the space must be linearly dependent
- A set of vectors is linearly independent if and only if the homogeneous linear system corresponding to the matrix of column vectors has only the trivial solution.
- A set of vectors is linearly independent if and only if the matrix of column vectors in reduced row-echelon form has every column as a pivot column.
- A subset of a linearly independent set is linearly independent.
- A vector can be written uniquely as a linear combination of vectors from independent subspaces.
- A vector can be written uniquely as a sum of a vector in a subspace and a vector orthogonal to the subspace.
- 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.
- Axioms of a vector space
- Bases
- Basic properties
- Basic properties of linear transformations
- Basic terminology
- Basic terminology and notation
- Block matrices
- Canonical forms of matrices
- Change of coordinates matrices are invertible
- Characteristic and minimal polynomials
- C^n is a vector space.
- Cofactors
- Composition
- Conjugating by a change of coordinates matrix converts matrix representations with respect to different bases.
- Conjugation
- Container for Linear Algebra
- Coordinates
- Coordinate vector spaces
- Cramer's rule
- Definition and terminology
- Definition of 0 matrix
- Definition of 0/trivial subspace
- Definition of 0 vector
- Definition of adjoint (conjugate transpose)
- Definition of adjugate/classical adjoint of a matrix
- Definition of (algebraic) multiplicity of an eigenvalue
- Definition of a lower triangular matrix
- Definition of angle between vectors
- Definition of an upper triangular matrix
- Definition of applying a polynomial to a linear transformation
- Definition of applying a polynomial to a square matrix
- Definition of augmented matrix (of a linear system)
- Definition of automorphism of a vector space
- Definition of a vector being orthogonal to a subspace
- Definition of band matrix
- Definition of basic/dependent/leading variable in a linear system
- Definition of basis of a vector space (or subspace)
- Definition of block diagonal matrix
- Definition of block/partitioned matrix
- Definition of change of coordinates matrix between two bases
- Definition of change-of-coordinates matrix relative to a given basis of R^n (or C^n)
- Definition of characteristic equation of a matrix
- Definition of characteristic polynomial of a linear transformation
- Definition of characteristic polynomial of a matrix
- Definition of Cholesky decomposition
- Definition of codomain of a linear transformation
- Definition of coefficient matrix of a linear system
- Definition of coefficients of a linear equation
- Definition of cofactor/submatrix of a matrix
- Definition of column rank of a matrix
- Definition of column space of a matrix
- Definition of column vector
- Definition of complement of a subspace
- Definition of composition of linear transformations
- Definition of conjugate of a matrix
- Definition of conjugate of a vector in C^n
- Definition of consistent linear system
- Definition of constant vector of a linear system
- Definition of coordinates relative to a given basis
- Definition of coordinate vector/mapping/representation relative to a given basis
- Definition of cross product
- Definition of determinant of a matrix as a cofactor expansion across the first row
- Definition of determinant of a matrix as a product of the diagonal entries in a non-scaled echelon form.
- Definition of diagonalizable linear transformation
- Definition of diagonalizable matrix
- Definition of diagonal matrix
- Definition of dimension of a vector space (or subspace)
- Definition of dimension of a vector space (or subspace) being finite or infinite
- Definition of direct sum of subspaces
- Definition of distance
- Definition of distance between vectors
- Definition of domain of a linear transformation
- Definition of echelon form of a linear system
- Definition of (echelon matrix/matrix in (row) echelon form)
- Definition of eigenspace of a linear transformation
- Definition of eigenspace of a matrix
- Definition of eigenvalue/characteristic value of a linear transformation
- Definition of eigenvalue of a matrix
- Definition of eigenvector/characteristic vector of a linear transformation
- Definition of eigenvector of a matrix
- Definition of elementary matrix
- Definition of entry/component of a vector
- Definition of equality of matrices
- Definition of equality of vectors
- Definition of equation operations on a linear system
- Definition of equivalent matrices
- Definition of equivalent systems of linear equations
- Definition of extended reduced row echelon form of a matrix
- Definition of free/independent variable in a linear system
- Definition of generalized inverse of a matrix
- Definition of generalized kernel/null space of linear transformation
- Definition of generalized range space of a linear transformation
- Definition of geometric multiplicity of an eigenvalue
- Definition of Gram-Schmidt process
- Definition of Hermitian/self-adjoint matrix
- Definition of Hessenberg form
- Definition of homogeneous linear system of equations
- Definition of how the action of a linear transformation on a basis extends to the whole space
- Definition of identity linear transformation
- Definition of identity matrix
- Definition of ill-conditioned linear system
- Definition of image (of a point) under a linear transformation
- Definition of inconsistent linear system
- Definition of independent subspaces
- Definition of index of nilpotency
- Definition of inner/dot product on C^n
- Definition of inner/dot product on R^n
- Definition of inner product
- Definition of inner product space
- Definition of intersection of subspaces
- Definition of invariant subspace of a linear transformation.
- Definition of inverse of a linear transformation
- Definition of inverse of a matrix
- Definition of invertible linear transformation
- Definition of invertible matrix
- Definition of invertible/nonsingular linear transformation
- Definition of isomorphic/isomorphism between vector spaces
- Definition of Jordan form
- Definition of kernel/null space of linear transformation
- Definition of kernel of linear transformation
- Definition of leading entry in a row of a matrix
- Definition of least-squares error of a linear system
- Definition of least-squares solution to a linear system
- Definition of left inverse of a matrix
- Definition of length/norm of a vector
- Definition of linear combination of vectors
- Definition of linear dependence relation on a set of vectors
- Definition of linear equation
- Definition of linearly dependent set of vectors: one of the vectors can be written as a linear combination of the other vectors
- Definition of linearly independent set of vectors: if a linear combination is 0
- Definition of linear transformation/homomorphism
- Definition of LU decomposition
- Definition of Markov matrix
- Definition of matrix
- Definition of matrix diagonalization
- Definition of matrix equation
- Definition of matrix in reduced row echelon form
- Definition of matrix multiplication
- Definition of matrix multiplication in terms of column vectors
- Definition of matrix null space (left)
- Definition of matrix null space (right)
- Definition of matrix representation of a linear system
- Definition of matrix representation of a linear transformation
- Definition of matrix representation of a linear transformation from a vector space to itself
- Definition of matrix representation of a linear transformation with respect to bases of the spaces
- Definition of matrix-scalar multiplication
- Definition of matrix-vector product
- Definition of m by n matrix
- Definition of minimal polynomial of a linear transformation
- Definition of minimal polynomial of a matrix
- Definition of multilinear function
- Definition of nilpotent linear transformation
- Definition of nilpotent matrix
- Definition of nonsingular matrix: matrix is invertible
- Definition of nonsingular matrix: the associated homogeneous linear system has only the trivial solution
- Definition of nontrivial solution to a homogeneous linear system of equations
- Definition of normal matrix
- Definition of norm/length of a vector
- Definition of (not necessarily orthogonal) projection onto a component of a direct sum
- Definition of nullity of a linear transformation
- Definition of nullity of a matrix
- Definition of one-to-one/injective linear transformation
- Definition of onto/surjective linear transformation
- Definition of orthogonal basis of a (sub)space
- Definition of orthogonal complement of a subspace
- Definition of orthogonally diagonalizable matrix
- Definition of orthogonal matrix
- Definition of (orthogonal) projection of one vector onto another vector
- Definition of (orthogonal) projection onto a subspace
- Definition of orthogonal set of vectors
- Definition of orthogonal subspaces
- Definition of orthogonal vectors
- Definition of orthonormal basis of a (sub)space
- Definition of orthonormal set of vectors
- Definition of parallel vectors
- Definition of permutation matrix
- Definition of pivot
- Definition of pivot column
- Definition of pivot position
- Definition of positive-definite matrix
- Definition of pre-image (of a point) under a linear transformation
- Definition of pre-image of linear transformation
- Definition of QR decomposition
- Definition of quadratic form
- Definition of range of a linear transformation
- Definition of range of linear transformation
- Definition of rank factorization of a matrix
- Definition of rank of a linear transformation
- Definition of rank of a matrix
- Definition of rational form
- Definition of reduced LU decomposition
- Definition of reduced row echelon form of a matrix
- Definition of right inverse of a matrix
- Definition of R^n (or C^n)
- Definition of (row) echelon form of a matrix
- Definition of row equivalent matrices
- Definition of row operations on a matrix
- Definition of row reduce a matrix
- Definition of row space of a matrix
- Definition of scalar
- Definition of scalar multiple of a linear transformation
- Definition of Schur triangulation
- Definition of similarity transform
- Definition of similar matrices
- Definition of singular matrix (not nonsingular)
- Definition of singular value decomposition (SVD)
- Definition of size of a matrix
- Definition of size of a vector
- Definition of skew-symmetric matrix
- Definition of solution set of a system of linear equations
- Definition of solution to a linear equation
- Definition of solution to a system of linear equations
- Definition of solution vector of a linear system
- Definition of spanning/generating set for a space or subspace
- Definition of span of a set of vectors
- Definition of square matrix
- Definition of subspace
- Definition of subspace spanned by a set of a set of vectors
- Definition of sum of linear transformations
- Definition of sum of matrices
- Definition of sum of subspaces
- Definition of symmetric matrix
- Definition of system of linear equations
- Definition of the 0/trivial subspace
- Definition of the determinant in terms of the effect of elementary row operations
- Definition of the imaginary part of a vector in C^n
- Definition of the least-squares linear fit to 2-dimensional data
- Definition of the (main) diagonal of a matrix
- Definition of the real part of a vector in C^n
- Definition of the standard basis of the m by n matrices
- Definition of the standard basis of the polynomials of degree at most n
- Definition of the standard matrix for a linear transformation
- Definition of the standard/natural basis of R^n (or C^n)
- Definition of trace of a matrix
- Definition of transpose of a matrix
- Definition of trivial linear dependence relation on a set of vectors
- Definition of trivial solution to a homogeneous linear system of equations
- Definition of unitary matrix
- Definition of unit matrix
- Definition of unit vector
- Definition of Vandermonde matrix
- Definition of vector
- Definition of vector addition
- Definition of vector-scalar multiplication
- Definition of vector sum/addition
- Definition of weights in a linear combination of vectors
- Description of a basis for the null space of a matrix from the reduced row-echelon form.
- Description of a spanning set for the null space of a matrix from the reduced row-echelon form.
- Description of the Gram-Schmidt process
- Determinants
- Determinants and operations on matrices
- Determinants axiomatically
- Determine if a particular set of vectors in R^3 in linearly independent
- Determine if a particular set of vectors spans R^3
- Determine if a particular vector is in the span of a set of vectors
- Determine if a particular vector is in the span of a set of vectors in R^2
- Determine if a particular vector is in the span of a set of vectors in R^3
- Dimension
- Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.
- Each vector can be written uniquely as a linear combination of vectors from a given basis.
- Echelon matrices
- Eigenspaces
- Eigenvalues and eigenvectors
- Eigenvalues and operations on matrices
- Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.
- Eigenvectors with distinct eigenvalues are linearly independent.
- Elementary matrices
- Elementary matrices are invertible/nonsingular.
- Equation operations on a linear system give an equivalent system.
- Equivalence theorem for injective linear transformations: The columns of the matrix of T are linearly independent.
- Equivalence theorem for injective linear transformations: The image of a basis for V is a basis for the range of T.
- Equivalence theorem for injective linear transformations: The inverse of T is a linear transformation on its range.
- Equivalence theorem for injective linear transformations: The kernel of T is 0.
- Equivalence theorem for injective linear transformations: The nullity of T is 0.
- Equivalence theorem for injective linear transformations: The null space of T is 0.
- Equivalence theorem for injective linear transformations: The rank of T is equals the number of columns in any matrix representation..
- Equivalence theorem for injective linear transformations: The rank of T is n.
- Equivalence theorem for injective linear transformations: T(x)=0 only for x=0.
- Equivalence theorem for nonsingular matrices: the columns of A are a basis for R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the columns of A are linearly independent.
- Equivalence theorem for nonsingular matrices: the columns of A span R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the determinant of A is nonzero.
- Equivalence theorem for nonsingular matrices: the dimension of the column space of A is n.
- Equivalence theorem for nonsingular matrices: the equation Ax=0 has only the trivial solution.
- Equivalence theorem for nonsingular matrices: the equation Ax=b has a solution for all b.
- Equivalence theorem for nonsingular matrices: the equation Ax=b has a unique solution for all b.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax has an inverse.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is an isomorphism.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is one-to-one/injective.
- Equivalence theorem for nonsingular matrices: the linear transformation given by T(x)=Ax is onto/surjective.
- Equivalence theorem for nonsingular matrices: the matrix A does not have 0 as an eigenvalue.
- Equivalence theorem for nonsingular matrices: the matrix A has a left inverse.
- Equivalence theorem for nonsingular matrices: the matrix A has an inverse.
- Equivalence theorem for nonsingular matrices: the matrix A has a right inverse.
- Equivalence theorem for nonsingular matrices: the matrix A has rank n.
- Equivalence theorem for nonsingular matrices: the matrix A is a change-of-basis matrix.
- Equivalence theorem for nonsingular matrices: the matrix A represents the identity map with respect to some pair of bases.
- Equivalence theorem for nonsingular matrices: the matrix A row-reduces to the identity matrix.
- Equivalence theorem for nonsingular matrices: the nullity of the matrix A is 0.
- Equivalence theorem for nonsingular matrices: the null space of the matrix A is {0}.
- Equivalence theorem for nonsingular matrices: there is a pivot position in every row of A.
- Equivalence theorem for nonsingular matrices: the rows of A are a basis for R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the rows of A are linearly independent.
- Equivalence theorem for nonsingular matrices: the rows of A span R^n (or C^n).
- Equivalence theorem for nonsingular matrices: the transpose of the matrix A has an inverse.
- Equivalence theorems for injective transformations
- Equivalent matrices represent the same linear transformation with resect to appropriate bases.
- Every basis for a vector space contains the same number of elements
- Every finite dimensional vector space over R (or C) is isomorphic to R^n (or C^n) for some n.
- Every matrix has an eigenvalue over the complex numbers.
- Every matrix is row-equivalent to a matrix in reduced row echelon form.
- Every matrix is row-equivalent to only one matrix in reduced row echelon form.
- Every nilpotent matrix is similar to one with 1 on subdiagonal blocks and all other entries 0.
- Every square matrix is conjugate
- Every square matrix is similar the sum of a diagonal and a nilpotent matrix.
- Every square matrix is similar to one in Jordan form.
- Example of a linear transformation on R^2: generic
- Example of a linear transformation on R^2: projection
- Example of a linear transformation on R^2: rotation
- Example of a linear transformation on R^2: shear
- Example of a linear transformation on R^3: rotation
- Example of a sum of vectors interpreted geometrically in R^2
- Example of (echelon matrix/matrix in (row) echelon form)
- Example of finding the inverse of a 2-by-2 matrix by row reducing the augmented matrix
- Example of finding the inverse of a 2-by-2 matrix by using a formula
- Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix
- Example of finding the inverse of a 3-by-3 matrix by using Cramer's rule
- Example of linear combination of vectors in R^2
- Example of matrix-vector product
- Example of multiplying 2x2 matrices
- Example of multiplying 3x3 matrices
- Example of multiplying matrices
- Example of multiplying nonsquare matrices
- Example of putting a matrix in echelon form
- Example of putting a matrix in echelon form and identifying the pivot columns
- Example of row reducing a 3-by-3 matrix
- Example of row reducing a 4-by-4 matrix
- Example of solving a 3-by-3 homogeneous matrix equation
- Example of solving a 3-by-3 homogeneous system of linear equations by row-reducing the augmented matrix
- Example of solving a 3-by-3 matrix equation
- Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix
- Example of using the echelon form to determine if a linear system is consistent.
- Example of vector-scalar multiplication in R^2
- Example of writing a given vector in R^3 as a linear combination of given vectors
- Examples
- Examples of vector spaces
- Factorization of matrices
- F^n is a vector space.
- For invertible linear transformations A and B
- For matrices
- Formula for computing the least squares solution to a linear system
- Formula for computing the least squares solution to a linear system.
- Formula for diagonalizing a real 2-by-2 matrix with a complex eigenvalue.
- Formula for the coordinates of a vector with respect to an orthogonal/orthonormal basis.
- Formula for the coordinates of the projection of a vector onto a subspace
- Formula for the determinant of a 2-by-2 matrix.
- Formula for the determinant of a 3-by-3 matrix.
- Formula for the inverse of a 2-by-2 matrix.
- Formula for the least-squares linear fit to 2-dimensional data
- Formula for the (orthogonal) projection of one vector onto another vector
- Formula for the spectral decomposition for a symmetric matrix
- For n-by-n invertible matrices A and B
- Gaussian elimination as a method to solve a linear system
- Gauss-Jordan procedure to put a matrix into reduced row echelon form
- Geometric description of span of a set of vectors in R^n (or C^n)
- Geometric picture of a 2-by-2 linear system
- Geometric picture of a 3-by-3 linear system
- Geometric picture of the solution set of a linear equation in 3 unknowns
- Geometric properties of linear transformations
- Geometric properties of linear transformations on R^2
- Geometric properties of R^n (or C^n)
- Hermitian matrices
- Hermitian matrices have real eigenvalues.
- Homogeneous linear systems are consistent.
- If A and B are n-by-n matrices
- If A is a matrix
- If a matrix has both a left and a right inverse
- If a set of vectors contains the 0 vector
- If a set of vectors in R^n (or C^n) contains more than n elements
- If a space is the direct sum of invariant subspaces
- If a square matrix has a one-sided inverse
- If a vector space has dimension n
- If B is a basis containing b and the b coordinate of c is nonzero
- If the product of a vector and a scalar is 0
- If two finite dimensional subspaces have the same dimension and one is contained in the other
- If two matrices have equal products with all vectors
- Inner products
- Inner products in coordinate spaces
- Inverse
- Isomorphic vector spaces have the same dimension.
- Isomorphism
- Least squares
- Linear algebra
- Linear combinations
- Linear (in)dependence
- Linear systems and echelon matrices
- Linear systems and matrices
- Linear systems have 0
- Linear systems of equations
- Linear transformations
- LU decomposition
- Matrices
- Matrices act as a transformation by multiplying vectors
- Matrices as linear transformations
- Matrix addition is commutative and associative.
- Matrix adjoint is an involution.
- Matrix conjugation is an involution.
- Matrix describing a rotation of the plane
- Matrix diagonalization
- Matrix equations
- Matrix equivalence
- Matrix inverse is an involution.
- Matrix inverses are unique: if A and B are square matrices
- Matrix multiplication can be viewed as the dot product of a row vector of column vectors with a column vector of row vectors
- Matrix multiplication is associative.
- Matrix multiplication is distributive over matrix addition.
- Matrix multiplication is not commutative in general.
- Matrix representation of a composition of linear transformations is given by a matrix product
- Matrix-scalar multiplication is commutative
- Matrix-scalar product is commutative
- Matrix transpose commutes with matrix inverse.
- Matrix transpose is an involution.
- Matrix-vector multiplication is a linear transformation.
- Matrix-vector product is associative
- Matrix-vector products
- Multiplication
- Multiplication by a change of coordinates matrix converts representations for different bases.
- Multiplication by a Hermitian matrix commutes with the standard inner product on C^n.
- Multiplication of block/partitioned matrices
- Multiplicity
- Multiplying a row by a scalar multiplies the determinant by that scalar.
- Nilpotent matrices
- Non-example of a linear transformation
- Nonsingular matrices and equivalences
- Normal matrices
- Norm and length
- Notation for entry of matrix
- Notation for the set of m by n matrices
- Operations on matrices
- Orthogonality
- Orthogonality and projection
- Parametric form of the solution set of a system of linear equations
- Parametric vector form of the solution set of a system of linear equations
- Particular types of matrices
- Projection
- Proof of several equivalences for nonsingular matrix
- QR decomposition
- Rank and mullity
- Rank and nullity
- Removing a linearly dependent vector from a set does not change the span of the set.
- R^n is a vector space.
- Row equivalence is an equivalence relation
- Row equivalent matrices have the same row space.
- Row equivalent matrices represent equivalent linear systems
- Row operations
- Row operations are given by multiplication by elementary matrices.
- Row operations do not necessarily preserve the column space.
- Scalar multiplication
- Similarity of matrices
- Similarity of matrices in an equivalence relation.
- Similar matrices have the same eigenvalues and the same characteristic polynomials.
- Spans
- Subspaces
- Subspaces associated to a linear transformation
- Subspaces associated to a matrix
- Switching two rows multiplies the determinant by -1.
- Symmetric matrices
- Symmetric matrices are square.
- Terminology
- The 0 scalar multiplied by any vector equals the 0 vector.
- The 0 vector is unique.
- The 0 vector multiplied by any scalar equals the 0 vector.
- The additive inverse of a vector equals the vector multiplied by -1.
- The additive inverse of a vector is called the negative of the vector.
- The additive inverse of a vector is unique.
- The adjoint of a matrix-scalar product is the product of the adjoint and the conjugate.
- The adjoint of a product of matrices is the product of the adjoints in reverse order.
- The adjoint of a sum is the sum of the adjoints.
- The Cauchy-Schwartz inequality
- The Cauchy-Schwarz inequality
- The Cayley-Hamilton theorem for a linear transformation
- The Cayley-Hamilton theorem for a matrix.
- The change of coordinates matrix between two bases exists and is unique
- The characteristic polynomial applied to the matrix gives the 0 matrix.
- The column space of a matrix is a vector space
- The column space of an m-by-n matrix is a subspace of R^m (or C^m)
- The composition of injective linear transformations is injective
- The composition of invertible linear transformations is invertible
- The composition of linear transformations is a linear transformation
- The composition of surjective linear transformations is surjective
- The condition number of matrix measures how close it is to being singular
- The conjugate of a matrix-scalar product is the product of the conjugates.
- The conjugate of a product of matrices is the product of the conjugates.
- The conjugate of a sum of vectors in C^n is the sum of the conjugates
- The conjugate of the sum of matrices is the sum of the conjugates.
- The conjugate of the transpose is the transpose of the conjugate.
- The conjugate of vector-scalar multiplication in C^n is the product of the conjugates.
- The coordinate vector relative to a given basis is a linear mapping to R^n (or C^n).
- The coordinate vector relative to a given basis is an injective linear mapping to R^n (or C^n).
- The coordinate vector relative to a given basis is a surjective linear mapping to R^n (or C^n).
- The crazy vector space is a vector space.
- The determinant function exists.
- The determinant function is unique.
- The determinant of a block diagonal matrix is the product of the determinants of the blocks.
- The determinant of a matrix can be computed as a cofactor expansion across any row.
- The determinant of a matrix can be computed as a cofactor expansion down any column.
- The determinant of a matrix can be expressed as a product of the diagonal entries in a non-scaled echelon form.
- The determinant of a matrix measures the area/volume of the parallelogram/parallelipiped determined by its columns.
- The determinant of a triangular matrix is the product of the entries on the diagonal.
- The determinant of the inverse of A is the reciprocal of the determinant of A.
- The determinant of the matrix of a linear transformation is the factor by which the area/volume changes.
- The dimension of a direct sum of subspaces is the sum of the dimensions of the subspaces.
- The dimension of a eigenspace is less than or equal to the (algebraic) multiplicity of the eigenvalue.
- The dimension of a subspace is less than or equal to the dimension of the whole space
- The dimension of the domain of an injective linear transformation is at most the dimension of the codomain.
- The dimension of the domain of a surjective linear transformation is at least the dimension of the codomain.
- The direct sum of a subspace and its orthogonal complement is the whole space.
- The echelon form can be used to determine if a linear system is consistent.
- The eigenspace of a linear transformation is a nontrivial subspace.
- The eigenvalues of a matrix are the roots/solutions of its characteristic polynomial/equation.
- The eigenvalues of a polynomial of a matrix are the polynomial of the eigenvalues.
- The eigenvalues of a power of a matrix are the power the eigenvalues.
- The eigenvalues of a scalar multiple of a matrix are the scalar multiples of the eigenvalues.
- The eigenvalues of a triangular matrix are the entries on the main diagonal.
- The eigenvalues of the inverse of a nonsingular matrix are the reciprocals of the eigenvalues.
- The eigenvectors of a normal matrix are an orthonormal basis.
- The geometry of linear systems
- The Gram-Schmidt process converts a linearly independent set into an orthogonal set.
- The identity matrix is the identity for matrix multiplication.
- The image of a linearly dependent set under a linear transformation is linearly dependent.
- The image of a linearly independent set under an injective linear transformation is linearly independent.
- The inner product of a vector with itself is the square of its norm/length.
- The intersection of subspaces is a subspace
- The inverse image of a subspace under a linear transformation is a subspace.
- The inverse of a linear transformation is a linear transformation
- The inverse of a matrix can be expressed in terms of its matrix of cofactors.
- The inverse of a matrix can be used to solve a linear system.
- The inverse of a matrix (if it exists) can be found by row reducing the matrix augmented by the identity matrix.
- The inverse of an invertible upper/lower triangular matrix is upper/lower triangular.
- The inverse of an isomorphism is an isomorphism.
- The inverse of a scalar multiple is the reciprocal times the inverse.
- The inverse of the inverse of a linear transformation is the original linear transformation
- The kernel/null space of a linear transformation is a subspace
- The kernels of powers of a linear transformation form an ascending chain
- The least squares solution to a linear system is unique if and only if the columns of the coefficient matrix are linearly independent.
- The left null space of a matrix is a subspace of R^m (or C^m).
- The matrix equation Ax=b has a solution if and only if b is a linear combination of the columns of A.
- The matrix representation of a composition of linear transformations is the product of the matrices.
- The matrix representation of a scalar multiple of linear transformations is the scalar multiple of the matrix.
- The matrix representation of a sum of linear transformations is the sum of the matrices.
- The matrix representation of the inverse of linear transformations is the inverse of the matricix.
- The minimal polynomial of a linear transformation exists and is unique.
- The minimal polynomial of a square matrix exists and is unique.
- The nonzero rows of an echelon form of a matrix are linearly independent.
- The nonzero rows of the reduced row-echelon form of a matris are a basis for the row space.
- The null space of a matrix is a subspace of R^n (or C^n).
- The null space of a matrix is the orthogonal complement of the column space.
- The number of pivots in the reduced row echelon form of a consistent system determines the number of free variables in the solution set.
- The number of pivots in the reduced row echelon form of a consistent system determines whether there is one or infinitely many solutions.
- The number of solutions to a linear system
- 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
- Theorem: a set of vectors is linearly independent if and only if whenever a linear combination is 0
- Theorem characterizing when a space is the direct sum of two subspaces
- Theorem describing matrix multiplication
- Theorem describing properties of the block matrices of the extended reduced row echelon form of a matrix
- Theorem describing spaces associated to the block matrices of the extended reduced row echelon form of a matrix
- Theorem describing the determinants of elementary matrices.
- Theorem describing the dimension of spaces associated to the block matrices of the extended reduced row echelon form of a matrix
- Theorem describing the vector form of sulutions to a linear system.
- The orthogonal complement of a subspace is a subspace.
- The (orthogonal) projection of a vector onto a subspace is the point in the subspace closest to the vector.
- The permutation expansion for determinants
- The pivot columns of a matrix are a basis for the column space.
- The preimage of a vector is a translation of the kernel of the linear transformation
- The product of square matrices is nonsingular if and only if each factor is nonsingular.
- The product of upper/lower triangular matrices is upper/lower triangular.
- The projection of a vector which is in a subspace is the vector itself.
- The QR decomposition of a nonsingular matrix exists.
- The range/image of a linear transformation is a subspace.
- The range of a linear transformation is a subspace
- The range spaces of powers of a linear transformation form a descending chain
- The rank of a matrix equals number of pivots in a reduced row echelon form.
- The rank of a matrix equals the rank of the linear transformation it represents.
- The rank plus the nullity of a linear transformation equals the dimension of the domain.
- The reduced row-echelon form of a matrix determines which subset of a spanning set is a basis.
- The row space and the column space of a matrix have the same dimension.
- The row space of a matrix is a vector space
- The set containing only 0 is a vector space.
- The set of all functions on a set is a vector space.
- The set of all polynomials is a vector space.
- The set of all polynomials of degree at most n is a vector space.
- The set of all sequences is a vector space.
- The set of linear transformations between two vector spaces is a vector space.
- The set of m by n matrices is a vector space.
- The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.
- The solutions to a homogeneous linear differential equation is a vector space.
- The solutions to a homogeneous system of linear equations is a vector space.
- The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.
- The span of a set of vectors is a subspace
- The spectral theorem for symmetric matrices
- The standard inner product of a vector with itself is 0 only for the 0 vector
- The standard inner product of a vector with itself is non-negative
- The standard inner product on C^n can be written as the product of a vector and the adjoint of a vector.
- The standard inner product on C^n commutes/anticommutes with scalar multiplication.
- The standard inner product on C^n is anticommutative.
- The standard inner product on R^n can be written as the product of a vector and the transpose of a vector.
- The standard inner product on R^n commutes with (real) scalar multiplication.
- The standard inner product on R^n is commutative.
- The standard inner product on R^n (or C^n) distributes over addition.
- The standard/natural basis of R^n (or C^n) is a basis.
- The sum of linear transformations is a linear transformation
- The sum of subspaces is a subspace
- The the image of a spanning set is a spanning set for the range space
- The transpose of a product of matrices is the product of the transposes in reverse order.
- The transpose of a sum of matrices is the sum of the transposes.
- The triangle inequality
- The union of bases from independent subspaces is a basis for the space.
- Transpose and adjoint
- Transpose commutes with scalar multiplication.
- Triangular matrices
- Two matrices of the same size are equivalent if and only if they have the same rank.
- Two vectors are orthogonal if and only if the Pythagorean Theorem holds.
- Unitary matrices
- Unitary matrices are invertible.
- Unitary matrices have orthogonal (orthonormal) rows/columns.
- Unitary matrices preserve inner products.
- Unitary matrices preserve orthogonal (orthonormal) bases.
- Using matrices to solve linear systems
- Vector space isomorphism is an equivalence relation.
- Vector spaces
- Vector spaces with the same dimension are isomprphic.
- Vector sum/addition interpreted geometrically in R^n (or C^n)
- Vector sum/addition is commutative and associative
- Visualise a linear transformation on R^2 by looking at the image of a region