0.1 Vector spaces

A finite dimensional vector space is the fundamental setting for matrix analysis.

0.1.1 Scalar field

Underlying a vector space is its field, or set of scalars. Typically $\mathbb{R}$ or $\mathbb{C}$, but it could be any set closed under addition and multiplication (referred to as $F$ when undefined). They must both be associative and commutative, each have an identity element, have existing inverses for all elements under addition and for all elements except the additive identity under multiplication; multiplication must be distributive over addition

0.1.2 Vector Spaces

A vector space $V$ over a field $F$ is a set $V$ of objects (vectors) that is closed under addition, is associative and commutative, and has an identity and additive inverses. Also closed under scalar multiplication.

0.1.3 Subspaces, span, and linear combinations

A subspace of a vector space $V$ over a field $F$ is a subset of $V$ that is, by itself, a vector space over $F$. A subset of $V$ is a subspace precisely when it is closed under vector addition and scalar multiplication.

For example, ${\left[\begin{array}{ccc}a& b& 0\end{array}\right]}^T:a,b\in \mathbb{R}$ is a subspace of ${\mathbb{R}}^3$.

An intersection of subspaces is always a subspace; a union is not always.

If $S$ is a subset of a vector space $V$ over a field $F$, span $S$ is the intersection of all subspaces of $V$ that contain $S$. If $S$ is nonempty, then span $S=\{a_1,v_1+....+a_k{v}_k:v_1,...,v_k\in S,a_1,...,a_k\in F$ and $k=1,2,...\}$. If $S$ is empty, then it is contained in every subspace of $V$. N

A linear combination of vectors in a vector space $V$ over a field $F$ is any expression of the form $a_1{v}_1+...+a_k{v}_k$ in which $k$ is positive and $a_1,...,a_k\in F$ and $v_1,...,v_k\in V$.

0.1.4 Linear dependence and linear independence

A list of vectors in $V$ are linearly dependent if and only if there are scalars $a_1,...,a_k\in F$, not all zero, such that $a_1{x}_1+...+a_k{x}_k=0$. Thus, a list of vectors $v_1,...,v_k$ is linearly dependent if and only if some nontrivial linear combination of $v_1,...,v_k$ is a zero vector. A list of vectors is said to have length $k$.

A list of two or more vectors is linearly dependent if one of the vectors is a linear combination of some of the others.

list of vectors is linearly independent if it is not linearly dependent.

For infinite lists, they are said to be linearly independent if some finite sublist is linearly independent.

The cardinality of a finite set is the number of its (necessarily distinct) elements. For a given list of vectors, the cardinality of the set is less than $k$ iff two or more vectors in the list are identical.

0.1.5 Basis

A linearly independent list of vectors whos span is $V$ is a basis for $V$. Each element of $V$ can be represented as a linear combination of vectors from the basis in one and only one way.

0.1.6 Extension to a basis

Any linearly independent list of vectors in $V$ may be extended to be a basis of $V$.

0.1.7 Dimension

The dimension of a vector space $V$ is defined by how long the list of vectors that is a basis to $V$ is. All basis for $V$ are of the same length.

The real vector space ${\mathbb{R}}^n$ has dimension $n$. The vector space ${\mathbb{C}}^n$ has dimension $n$ over the field $\mathbb{C}$ but over the field $\mathbb{R}$ it has dimension $2n$.

0.1.8 Isomorphism

If $U$ and $V$ are vector spaces over the same scalar field $F$, and if $f:U\to V$ is an invertible function for all $x,y\in U$ and all $a,b\in F$, then $f$ is said to be an isomorphism and $U$ and $V$ are said to be isomorphic.