# Index
[[2 dimensional Hilbert space]]
[[Adjoint]]
[[Affine space]]
[[Anti-Hermitian operator]]
[[Anti-linear map]]
[[Bilinear map]]
[[Block matrices]]
[[Cauchy-Schwarz inequality]]
[[Change of basis]]
[[Complex matrix exponentials]]
[[Complex vector spaces]]
[[Cramer's rule]]
[[Cross products]]
[[Complete sets of compatible operators]]
[[Commuting operators]]
[[Degenerate eigenvalues]]
[[Determinants]]
[[Diagonalizable matrices]]
[[Direct sums of vector spaces]]
[[Eigenvalues and eigenvectors]]
[Eigenvalues](Eigenvalues.md)
[Eigenvectors](Eigenvectors)
[[Euclidean space]]
[[Finite dimensional vector spaces]]
[Fock Space](Fock%20Space.md)
[[Functions of linear operators]]
[[Generalized eigenvectors]]
[Gram-Schmidt Orthonormalization](Gram-Schmidt%20orthonormalization)
[Hausdorff pre-Hilbert Space](Hausdorff%20pre-Hilbert%20Space.md)
[[Hermitian operators]]
[[Hilbert Space]]
[[Hilbert Space dimension]]
[[Idempotent operators]]
[[Inner products]]
[[Invariant subspaces]]
[[Involutive operators]]
[linear basis](linear%20basis)
[[Linear isometry]]
[[Linear map]]
[Linear span](Linear%20span.md)
[[linear operator]]
[[Matrices]]
[[Matrix exponentials]]
[[Matrix similarity]]
[[Matrix logarithms]]
[[Finite dimensional Hilbert spaces]]
[[Vector normalization]]
[[Normal operators]]
[[Null space]]
[[Operator inverse]]
[Orthogonal complement of a subspace](Orthogonal%20complement%20of%20a%20subspace)
[[Orthogonal operators]]
[[Orthonormal bases]]
[[Orthonormal vectors]]
[[Outer product]]
[overcomplete basis](overcomplete%20basis)
[[Partial trace]]
[Pauli Matrices](Pauli%20Matrices.md)
[[Pfaffians]]
[[Perron-Frobenius theorem]]
[Positive semidefinite operators](Positive%20semidefinite%20operators.md)
[[Projection]]
[[Range of a linear operator]]
[[Rayleigh quotient]]
[[Real vector spaces]]
[[Singular value decomposition]]
[[Spectral Theorem]]
[[Sums of vector spaces]]
[[Sylvester's matrix theorem]]
[[Topological vector spaces]]
[[Trace]]
[[Transpose of a linear map]]
[[Triangle inequality]]
[[unbounded]]
[[Unitary operators]]
[[vector norm]]
[[Vector outer products]]
[[Vector spaces]]
[Vector space dimension](Vector%20space%20dimension.md)
## Sub-Indices
[Tensors and Multilinear Algebra (index)](Tensors%20and%20Multilinear%20Algebra%20(index).md)
[Vector analysis (index)](Vector%20analysis%20(index).md)
[Functional analysis (index)](Functional%20analysis%20(index).md)
---
# Proofs and examples
[[Proof of the Cauchy-Schwarz inequality on complex vector spaces]]
[[Proof of the completeness relation for orthornormal bases]]
[[Proof of the triangle inequality on complex vector spaces]]
[[Proof of equivalence of definitions of Hermiticity]]
[[Proof that Hermitian operators have real eigenvalues]]
[[Proof that eigenvectors of Hermitian operators are orthogonal]]
[[Proof that the sum of subspaces is also a subspace]]
---
# Basic Concepts
## Vectors
Vectors are [lists](n-tuples.md) of elements (to start, here we only consider vectors containing numbers) either written as columns or rows (though the first encounter with vectors is through the introduction of column vectors). The dimensionality of a vector is determined by how many elements it has. Consider here a vector $\mathbf{v}$ containing only real numbers. Thus we define
$\mathbf{v}=\begin{pmatrix}x^1\\ \vdots\\ x^m \end{pmatrix}$
as a column vector and
$\mathbf{v}^T=\begin{pmatrix}x_1 ... x_n \end{pmatrix}$
as a row vector where if we view a vector as being a $1 \times n$ [matrix](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Matrices) we take the [transpose](Transpose%20of%20a%20linear%20map.md) to switch between writing the vector as a row or a column.
Note that the column vector indices are not exponents. We choose this notation because it is very useful in differentiating between elements of [vector spaces](Vector%20spaces.md) and [dual spaces](Vector%20spaces.md#Dual%20Spaces) in [multilinear algebra](Tensors%20and%20Multilinear%20Algebra%20(index).md).
### Vector notation
When representing a vector in a definition or equation, I use the following conventions:
* vectors in unspecified and [[Euclidean space]]s: $\mathbf{v}$.
* [complex](Complex%20vector%20spaces.md) vectors: bra-ket notation, $|v\rangle$ and $\langle v|$ in the corresponding [dual space](Vector%20spaces.md#Dual%20Spaces). and the [complex inner product](Inner%20products.md#Complex%20inner%20products) is written as $\langle u,v \rangle.$
* For [complex vectors](Complex%20vector%20spaces.md) in [Hilbert spaces](Hilbert%20Space.md) we will write the inner product as $\langle u|v \rangle,$ which follows from conventions used in quantum mechanics.
### The 0 vector
One may write a vector whose elements are all $0$ and this will be denoted as $\mathbf{0}.$
### Vector Spaces
Sets of vectors may form what we refer to as [vector spaces.](Vector%20spaces.md)
### Vector length
#### Unit Vectors
### vector addition
#### linear combinations
A _linear combination_ of a list of vectors, $(v_1,...,v_m)$ is a [sum of vectors](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#vector%20addition) of the form
$a_1v_1+...+a_mv_m$
where $a_1,...,a_m \in \mathbb{F}$ (that is, where $a_1,...,a_m$ are numbers.)
%%What is a list here? as in, why that word? not set?%%
### vector multiplication
There are two types of [vector](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) multiplication. One of them referred to as the _dot product_ in $\mathbb{R}^n,$ which generalizes to to the so called [inner product](Inner%20products.md) in other number fields (e.g. $\mathbb{C}^n$) and the [cross product.](Cross%20products.md)
#### dot product
For a pair of [vectors](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors), $\mathbf{x}$, $\mathbf{y}$ in $\mathbb{R}^n,$ (each vector contains $n$ elements) we define the inner product as
$\mathbf{x}\cdot\mathbf{y}=x_1y_1+x_2y_2+...x_ny_n=\sum_j^n x_j y_j$
This is equivalent to the _[real inner product](Inner%20products.md#Real%20inner%20products) in $\mathbb{R}^n$,_ which is defined as
#### Cross products
### Orthogonal vectors
A pair of vectors [vectors,](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) $\mathbf{u}$ and $\mathbf{v}$ where $\mathbf{u}\neq\mathbf{v}$ are _orthogonal_ if their [inner product](Inner%20products.md) is 0. We denote a pair of orthogonal vectors as $\mathbf{u}\perp\mathbf{v}$^566641
%%These needs to be connected to geometry somehow%%
### Parallel vectors
We denote a pair of _parallel_ [vectors,](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) $\mathbf{u}$ and $\mathbf{v}$ as $\mathbf{u}\parallel\mathbf{v}$
Two vectors, $\mathbf{u}$ and $\mathbf{v}$, in a [three dimensional Euclidean space](Geometry%20(index).md#3D%20Euclidean%20space) are parallel if $\mathbf{u}\times\mathbf{v} = 0,$ where $\times$ denotes the [cross product.](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#cross%20product)
## Transformations and operators
%%Here we describe the distinction between a transformation and an operator and that operators are what give rise to transformations%%
### Operators
%%Here specify the idea that operators can be represented as matrices and that other operators cannot be.%%
#### Matrices
A [matrix](Matrices.md) is a way of specifying a [linear map](Linear%20map.md) using a two-dimensional array of numbers. A matrix, $A$ emerges as follows from a system of linear equations that are arranged in a [vector](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) where the coefficients in the linear equations $A^i_j$, are also the _[matrix elements](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#matrix%20element%20notation)_. ^fc2a1f
$\begin{pmatrix}A^1_1x^1 + ... + A^1_mx^n\\ \vdots\\ A^m_1x^1 + ... + A^m_nx^n \end{pmatrix}=A\begin{pmatrix}x^1\\ \vdots\\ x^m \end{pmatrix}=\begin{pmatrix}A^1_1 & ... & A^1_m\\ \vdots & & \vdots \\ A^m_1 & ... & A^m_n \end{pmatrix}\begin{pmatrix}x^1\\ \vdots\\ x^m \end{pmatrix}$ ^c93b25
In linear algebra we examine the properties that emerge from this for different sorts of arrangements of matrix elements. More details on the algebraic properties of matrices can be found [here](Matrices.md). In addition, a lot of what we find to be true for finite matrices directly generalizes to [operators](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Operators) which would in principle be infinitely sized if we were to try to represent them as matrices.
#### matrix element notation
Matrix elements are denoted by subscript or superscript indices indicating the row ($m$) and column ($n$). This means that for a matrix $A,$ its elements are denoted as
$A^m_n$
or in contexts where whether an index is a superscript or subscript doesn't matter, we may write
$A_{mn}$
In practice, rather than writing out whole matrices in a calculation meant to show some set of linear relations, we may show it in terms of individual _[matrix](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Matrices) elements_.
%%Specify what context the positions of the scripts don't matter%%
##### Matrix addition
##### Matrix multiplication
In terms of [matrix elements](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#matrix%20element%20notation) the product of two matrices $C=AB$ may be expressed as
$C_{m}^l=(AB)_m^l = \sum_n A_m^n B_n^l$
or as
$C_{ml}=(AB)_{ml} = \sum_n A_{mn} B_{nl}$
in contexts where whether the indices are lowered or raised doesn't matter to the calculation. The parentheses indicate that the subscript corresponds with the product of $A$ and $B.$
##### Superposition principle
##### Matrix Decomposition
---
# Bibliography
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[Axler, S., Down With Determinants, _American Mathematical Monthly 102_, 1995.](Axler,%20S.,%20Down%20With%20Determinants,%20American%20Mathematical%20Monthly%20102,%201995..md)
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[Cherney D., Denton T., and Waldron A., _Linear Algebra_, 2013](Cherney%20D.,%20Denton%20T.,%20and%20Waldron%20A.,%20Linear%20Algebra,%202013..md)
[Griffiths D. J., _Introduction to Quantum Mechanics_, Pearson Prentice Hall, 2nd edition, 2005.](Griffiths%20D.%20J.,%20Introduction%20to%20Quantum%20Mechanics,%20Pearson%20Prentice%20Hall,%202nd%20edition,%202005..md)
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[joshphysics (https://physics.stackexchange.com/users/19976/joshphysics), Trace of an operator matrix (Quantum computation and quantum information), URL (version: 2015-04-24): https://physics.stackexchange.com/q/104155](joshphysics%20Trace%20of%20an%20operator%20matrix%20(Quantum%20computation%20and%20quantum%20information),%20URL%20(version%202015-04-24).md)
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[von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md)
[Woit, Peter. _Quantum Theory, Groups and Representations: An Introduction_, Springer, 2017](Woit,%20Peter.%20Quantum%20Theory,%20Groups%20and%20Representations%20An%20Introduction,%20Springer,%202017.md)
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/VectorSpaces
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Operators/Matrices
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/MultiLinearAlgebra
#MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/MultiLinearAlgebra/Tensors
#Bibliography
#index