# Category:Linear Algebra

This category contains results about Linear Algebra.

Definitions specific to this category can be found in Definitions/Linear Algebra.

**Linear algebra** is the branch of algebra which studies vector spaces and linear transformations between them.

## Subcategories

This category has the following 32 subcategories, out of 32 total.

### A

### B

### C

### D

### F

### H

### L

### M

### N

### O

### Q

### R

### S

### T

### V

### Z

## Pages in category "Linear Algebra"

The following 123 pages are in this category, out of 123 total.

### C

- Cardinality of Generator No Less than Dimension
- Cardinality of Linearly Independent Set No Greater than Dimension
- Cauchy-Bunyakovsky-Schwarz Inequality
- Cayley-Hamilton Theorem/Matrices
- Cayley-Menger Determinant
- Characterization of Left Null Space
- Complex Numbers form Vector Space
- Composition of Linear Real Functions
- Condition for Composition of Linear Real Functions to be Commutative
- Condition for Planes to be Parallel
- Condition for Straight Lines in Plane to be Parallel
- Conditions for Homogeneity
- Conditions for Homogeneity/Plane
- Conditions for Homogeneity/Straight Line
- Conjugate Transpose is Involution

### D

- Differentiation on Polynomials is Linear Operator
- Dimension of Finitely Generated Vector Space is Well-Defined
- Dimension of Proper Subspace Less Than its Superspace
- Dimension of Vector Space is Well-Defined
- Direct Product of Vector Spaces is Vector Space
- Division Ring is Vector Space over Prime Subfield

### E

- Elementary Row Operations by Matrix Multiplication
- Elementary Row Operations by Matrix Multiplication/Corollary
- Empty Set is Linearly Independent
- Equation of Plane
- Equation of Straight Line in Plane
- Equivalence of Definitions of Change of Basis Matrix
- Equivalent Matrices have Equal Rank
- Expression of Vector as Linear Combination from Basis is Unique
- Expression of Vector as Linear Combination from Basis is Unique/General Result

### F

### G

### H

- Hermitian Matrix has Real Eigenvalues/Corollary
- Hermitian Operators have Orthogonal Eigenvectors
- Hermitian Operators have Real Eigenvalues
- Hermitian Operators have Real Eigenvalues and Orthogonal Eigenvectors
- Homogeneous Linear Equations with More Unknowns than Equations
- Homogeneous System has Zero Vector as Solution
- Homogeneous System has Zero Vector as Solution/Corollary
- Homomorphic Image of Vector Space

### I

### L

- Linear Combination of Sequence is Linear Combination of Set
- Linear Function on Real Numbers is Bijection
- Linear Operator on General Logarithm
- Linear Operator on the Plane
- Linear Transformation of Vector Space Equivalent Statements
- Linear Transformation of Vector Space Monomorphism
- Linear Transformations Isomorphic to Matrix Space
- Linear Transformations Isomorphic to Matrix Space/Corollary
- Linearly Dependent Sequence of Vector Space
- Linearly Independent Subset also Independent in Generated Subspace
- Lines are Subspaces of Plane

### N

### R

- Rank and Nullity of Transpose
- Rank is Dimension of Subspace
- Rank Plus Nullity Theorem
- Rational Numbers form Vector Space
- Real Linear Subspace Contains Zero Vector
- Real Numbers form Vector Space
- Real Symmetric Matrix is Hermitian
- Real Vector Space is Vector Space
- Reflection of Plane in Line through Origin is Linear Operator
- Results Concerning Annihilator of Vector Subspace
- Results concerning Generators and Bases of Vector Spaces
- Ring of Endomorphisms
- Row Equivalent Matrix for Homogeneous System has same Solutions
- Row Equivalent Matrix for Homogeneous System has same Solutions/Corollary

### S

- Same Dimensional Vector Spaces are Isomorphic
- Similarity Mapping of Plane is Linear Operator
- Simultaneous Equation With Two Unknowns
- Singleton is Linearly Independent
- Solution to Simultaneous Linear Equations
- Standard Ordered Basis is Basis
- Stretching and Contraction Mappings of Plane are Linear Operators
- Subset of Linearly Independent Set is Linearly Independent
- Subset of Module Containing Identity is Linearly Dependent
- Subspaces of Dimension 2 Real Vector Space
- Sufficient Conditions for Basis of Finite Dimensional Vector Space
- Superset of Linearly Dependent Set

### T

### V

- Vector Space has Basis
- Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set
- Vector Space has Basis between Linearly Independent Set and Spanning Set
- Vector Space has Basis/Corollary
- Vector Space has Unique Additive Identity
- Vector Space has Unique Additive Inverses
- Vector Space of All Mappings
- Vector Space on Cartesian Product
- Vector Space over Division Subring
- Vector Space over Subring
- Vector Subspace Dimension One Less
- Vector Subspace of Real Vector Space
- Vectorization of Product of Three Matrices