# Category:Linear Algebra

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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 65 subcategories, out of 65 total.

### A

### B

- Boolean Prime Ideal Theorem (22 P)

### C

### D

- Dimension of Proper Subspace (1 P)
- Direct Sums (1 P)
- Discriminants (empty)

### E

- Examples of Linear Equation (2 P)

### F

- Floquet's Theorem (3 P)

### G

- Generators of Modules (1 P)
- Grassmann's Identity (3 P)

### H

### J

- Jordan Canonical Form (1 P)

### L

- Linear Combinations (empty)
- Linear Forms (empty)

### M

- Matrix Spaces (1 P)

### N

### O

- Operator Theory (7 P)
- Orthonormal Bases of Vector Spaces (empty)

### Q

### R

- Representation Theory (11 P)

### S

- Scalar Multiplication (7 P)
- Similarity Mappings (6 P)
- Standard Ordered Bases (4 P)

### T

- Tensor Theory (empty)

### U

### V

- Vector Subspaces (17 P)

### Z

## Pages in category "Linear Algebra"

The following 101 pages are in this category, out of 101 total.

### C

- Cardinality of Linearly Independent Set is No Greater than Dimension
- Cauchy-Bunyakovsky-Schwarz Inequality
- Cayley-Hamilton Theorem/Matrices
- Change of Basis Matrix under Linear Transformation
- Characterization of Left Null Space
- Complex Numbers form Vector Space over Reals
- Complex Numbers form Vector Space over Themselves
- Composition of Linear Real Functions
- Condition for Composition of Linear Real Functions to be Commutative
- Condition for Planes to be Parallel
- Conditions for Homogeneity
- Conditions for Homogeneity/Plane
- Conditions for Homogeneity/Straight Line
- Conjugate Transpose is Involution
- Content of Cayley-Menger Determinant
- Cramer's Rule

### D

### E

- Eigenvalues of Hermitian Operator have Orthogonal Eigenspaces
- Empty Set is Linearly Independent
- Equation of Plane
- Equation of Straight Line in Plane
- Equivalence of Definitions of Change of Basis Matrix
- Equivalent Statements for Vector Subspace Dimension One Less
- Existence of Minimal Polynomial for Square Matrix over Field
- Existence of Ordered Dual Basis
- Existence of Scalar for Vector Subspace Dimension One Less
- Expression of Vector as Linear Combination from Basis is Unique
- Expression of Vector as Linear Combination from Basis is Unique/General Result

### 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 Transformation of Vector Space Equivalent Statements
- 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 Plus Nullity Theorem
- Rational Numbers form Vector Space
- Real Linear Subspace Contains Zero Vector
- Real Numbers form Vector Space
- Real Symmetric Matrix is Hermitian
- Results concerning Generators and Bases of Vector Spaces
- 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
- Sample Matrix Independence Test
- Sample Matrix Independence Test/Examples
- Simultaneous Equation With Two Unknowns
- Singleton is Linearly Independent
- Subset of Linearly Independent Set is Linearly Independent
- Subset of Module Containing Identity is Linearly Dependent
- Subspaces of Dimension 2 Real Vector Space
- Superset of Linearly Dependent Set is Linearly Dependent