| parent,child |
| Linear Algebra,System of Linear Equations |
| Linear Algebra,Matrices |
| Linear Algebra,Determinants |
| Linear Algebra,Subspace of R^n |
| Linear Algebra,Vector Space |
| Linear Algebra,Linear Transformation |
| Linear Algebra,Eigenvalue and Diagonalization |
| System of Linear Equations,System of Linear Equations extended |
| System of Linear Equations extended,coefficient matrix |
| System of Linear Equations extended,augmented matrix |
| System of Linear Equations extended,Solution Set |
| Solution Set,Solution Consistency |
| System of Linear Equations,Elementary row operations |
| Elementary row operations,Equivalent linear systems |
| Equivalent linear systems,Solution Set |
| Elementary row operations,Row Echelon Form |
| Row Echelon Form,Reduced row echelon form |
| Reduced row echelon form,Pivot |
| System of Linear Equations,Gauss -Jordan Method |
| Gauss -Jordan Method,basic variable |
| Gauss -Jordan Method,free variables |
| Gauss -Jordan Method,General solution of the system |
| Gauss -Jordan Method,Existence and Uniqueness Theorem |
| System of Linear Equations,Homogeneous System |
| Homogeneous System,Trivial Solution |
| Homogeneous System,Nontrivial Solution |
| Homogeneous System,Properties of Homogeneous System |
| Homogeneous System,Parametric Vector Form |
| Matrices,Basic Matrix Operations |
| Basic Matrix Operations,Addition and Scalar Multiplication |
| Addition and Scalar Multiplication,Addition/Scalar Mult. Properties |
| Addition and Scalar Multiplication,Matrix Multiplication |
| Matrix Multiplication,Properties for Matrix Multiplication |
| Matrix Multiplication,Power of Matrix |
| Basic Matrix Operations,Identity Matrix |
| Basic Matrix Operations,Transpose of A |
| Transpose of A,Properties for Transpose |
| Matrices,Partitioned Matrices |
| Partitioned Matrices,Blocks (Submatrices) |
| Partitioned Matrices,Partitioned Matrix Operations |
| Partitioned Matrices,Block Multiplication |
| Matrices,Nonsingular Matrices |
| Nonsingular Matrices,Invertible |
| Nonsingular Matrices,Inverse of matrix |
| Nonsingular Matrices,Singular Matrix |
| Nonsingular Matrices,Properties for Nonsingular Matrices |
| Nonsingular Matrices,Nonsingular Matrix Conditions |
| Nonsingular Matrices,Algorithm for computing the inverse |
| Nonsingular Matrices,Inverse Formula |
| Determinants,Determinant Extended |
| Determinant Extended,Determinant for 2 x 2 matrices |
| Determinant Extended,Determinant of 3 x 3 matrices |
| Determinant Extended,Cofactor |
| Cofactor,computing determinant by cofactor |
| Determinant Extended,Triangular and diagonal matrix |
| Triangular and diagonal matrix,Upper or Lower triangular Matrix |
| Triangular and diagonal matrix,Determinant of triangular matrices |
| Determinant Extended,Row of Zeros / Column of Zeros |
| Determinant Extended,Determinant for transpose |
| Determinants,Elementary Operations for Determinants |
| Elementary Operations for Determinants,Elementary Row Op - Scaling |
| Elementary Operations for Determinants,Elementary Row Op - Replacement |
| Elementary Operations for Determinants,Elementary Row Op - Interchange |
| Determinants,Properties of Determinants |
| Properties of Determinants,Determinant for product |
| Properties of Determinants,Determinant for scalar multiple |
| Properties of Determinants,Determinant for inverse |
| Determinants,Adjoint (Adjugate) |
| Adjoint (Adjugate),Formula for adjoint |
| Determinants,Nonsingular Matrices |
| Determinants,Cramer's rule |
| Subspace of R^n,Linear Combination |
| Linear Combination,Vectors |
| Vectors,Algebraic Properties of R^n |
| Linear Combination,Weights |
| Linear Combination,Spanning Set |
| Spanning Set,Spanning R^m |
| Spanning Set,Equivalent Conditions for Spanning Set |
| Spanning Set,Subspace |
| Subspace of R^n,Linear Independence |
| Linear Independence,Linearly Independent |
| Linear Independence,Linearly dependent |
| Linear Independence,Facts about linear independence |
| Subspace of R^n,Subspace |
| Subspace,Properties for Subspace |
| Subspace,Vector Space and Subspace |
| Subspace of R^n,Nullspace and Columnspace |
| Nullspace and Columnspace,Column Space |
| Column Space,Subspace |
| Nullspace and Columnspace,Null Space |
| Null Space,Subspace |
| Nullspace and Columnspace,Basis for null space |
| Nullspace and Columnspace,Basis for column Space |
| Subspace of R^n,Basis and Dimension |
| Basis and Dimension,Basis for a Subspace |
| Basis and Dimension,Dimension of a Subspace |
| Basis and Dimension,The Spanning set theorem |
| Basis and Dimension,Basis and Invertible Matrix |
| Basis and Invertible Matrix,Invertible |
| Basis and Dimension,The basis theorem |
| The basis theorem,Number of vectors in different bases |
| Basis and Dimension,Basis and linearly dependent |
| Basis and linearly dependent,Linearly dependent |
| Basis and Dimension,Dimension of Nul A and Col A |
| Dimension of Nul A and Col A,Dimension of Col A |
| Dimension of Nul A and Col A,Dimension of Nul A |
| Subspace of R^n,Row Space |
| Row Space,Row Space and Row Equivalence |
| Subspace of R^n,Rank |
| Rank,Dimension of Col A |
| Rank,The rank theorem |
| Rank,Invertible Matrix Conditions |
| Vector Space,Vector Space and Subspace |
| Vector Space,Linear Combination |
| Vector Space,Linear Independence |
| Vector Space,Basis and Dimension |
| Vector Space,Basis for a vector space |
| Vector Space,Dimension of a vector space |
| Vector Space,Change of Basis |
| Change of Basis,B -Coordinate Vector |
| Change of Basis,Coord. Vector Independence |
| Change of Basis,Change -of -Coordinates in R^n |
| Change -of -Coordinates in R^n,Change -of -Coordinates Matrix |
| Linear Transformation,Linear Transformation Extended |
| Linear Transformation Extended,Transformation |
| Transformation,Domain |
| Transformation,Codomain |
| Transformation,Image |
| Transformation,Range |
| Linear Transformation Extended,Linear Transformation Definition |
| Linear Transformation Extended,LT Equivalent Condition |
| Linear Transformation Extended,Properties of linear transformations |
| Linear Transformation,LT Matrix Representation |
| LT Matrix Representation,LT Matrix (R^n to R^m) |
| LT Matrix Representation,LT Matrix (V to W) |
| Linear Transformation,Kernel and range |
| Kernel and range,Kernel |
| Kernel and range,Range |
| Kernel and range,Dimension of kernel and image |
| Linear Transformation,Injective and Surjective |
| Injective and Surjective,Injective |
| Injective,Injective LT Conditions |
| Injective and Surjective,Surjective |
| Surjective,Surjective LT Conditions |
| Injective and Surjective,Bijective |
| Bijective,Bijective LT Conditions |
| Injective and Surjective,One -to -One vs Onto |
| Injective and Surjective,Standard Matrix (Diff. Bases) |
| Eigenvalue and Diagonalization,Eigenvalues and Eigenvectors |
| Eigenvalues and Eigenvectors,Eigenvalues and Eigenvectors extended |
| Eigenvalues and Eigenvectors extended,Complex Eigenvalues and eigenvectors |
| Eigenvalues and Eigenvectors,Characteristic Equation |
| Eigenvalues and Eigenvectors,Eigenspace |
| Eigenvalues and Eigenvectors,Eigenvalues and Diagonal Entries |
| Eigenvalues and Diagonal Entries,Trace of Matrix |
| Eigenvalues and Eigenvectors,Eigenvalues of Triangular Matrix |
| Eigenvalues and Eigenvectors,Algebraic Multiplicity |
| Eigenvalues and Eigenvectors,Geometric Multiplicity |
| Eigenvalue and Diagonalization,Similarity |
| Similarity,Eigenvalues and Similarity |
| Eigenvalue and Diagonalization,Diagonalization |
| Diagonalization,Diagonal Matrix |
| Diagonalization,Diagonalization Theorem |
| Diagonalization,Linearly Independent eigenvectors |
| Diagonalization,Distinct Eigenvalues and Diagonalizable |
| Diagonalization,Factorization with real Matrices |
| Diagonalization,Standard Matrix (Diff. Bases) |
| Eigenvalue and Diagonalization,Jordan Canonical Form |
|
|
