Vector space

Vector spaces, also known as linear spaces, are an abstract mathematical construct with many important applications in the natural sciences, in particular in physics and numerous areas of mathematics. Some vector spaces make sense somewhat intuitively, such as the space of 3D vectors in standard Euclidean space, and the language that we use when talking about these intuitive spaces has been taken to describe the more abstract notion as well. For example, we know how to add vectors and multiply them by real numbers (scalars) in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} , and these notions of vector addition and scalar multiplication are defined in a more general sense (as we will see below).

Vector spaces are important because many different mathematical objects that at first glance seem unrelated in fact share a common structure. By defining this structure and proving things about it in general, we are then able to apply these results to each specific case without having to re-prove them each time. Besides vectors in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^3} that are relatively easy to visualize, we can make a vector space out of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} for any natural number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ; or the complex plane or powers of it, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}^n} ; or polynomials of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .

No matter what vector space you have to work with though, it is often useful to keep a picture of either 2D or 3D space in mind. This helps when thinking of things such as orthogonal polynomials or matrices.

Definition

A vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} over a field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is a set that satisfies certain axioms (see below) and which is equipped with two operations, vector addition and scalar multiplication. Vector addition is defined as a map

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +:\quad V\times V\to V}

that takes the ordered pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\vec{u},\vec{v})\in V\times V} to the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}+\vec{v}} . Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times} represents the Cartesian product between sets. Scalar multiplication is defined in a similar way, as a map

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cdot:\quad F\times V\to V}

that takes the ordered pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,\vec{u})\in F\times V} to the vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\cdot\vec{u}} . Note that frequently the dot representing scalar multiplication is omitted, the result being written simply as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\vec{u}} instead. This is especially common when an inner product will also be defined on the vector space, with the dot then representing the inner product between two vectors. It is important to keep in mind the distinction between scalar multiplication, which multiplies one vector by a scalar, and an inner or scalar product, that combined two vectors to yield a scalar.

Axioms of a vector space

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a set, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{v}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{w}} elements of that set, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} scalar elements of a field, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a vector space if the following axioms hold true for all choices of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u},\ \vec{v},\ a,\ b}

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is closed under addition

The vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}+\vec{v}} is also an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . This is automatically satisfied when the addition operation is defined as being injective as it was above. Care must be taken however if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is a subset of some larger set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +:\,\,V\times V\to W} , as is often the case when looking at subspaces.

2. Addition is commutative

The order in which two vectors are added does not affect the result, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}+\vec{v}=\vec{v}+\vec{u}} .

3. Addition is associative

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}+(\vec{v}+\vec{w})=(\vec{u}+\vec{v})+\vec{w}} . This means that even though addition is strictly defined as a binary operation, the object Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{u}+\vec{v}+\vec{w}} is well defined.

4. An additive identity exists in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}

Labeled Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{0}} , the additive identity or zero vector satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{0}+\vec{u}=\vec{u}+\vec{0}=\vec{u}} .

5. The additive inverse exists in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V}

A vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\vec{u}} can be found such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\vec{u}+\vec{u}=\vec{u}+(-\vec{u})=\vec{0}} .

6. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is closed under scalar multiplication

The vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\vec{u}} is itself an element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} .

7. Scalar multiplication is distributive over addition in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a+b)\vec{u}=a\vec{u}+b\vec{u}} . It is important to note that the addition occurring on the left-hand side of this equality is a 'different operation' from the addition on the right-hand side. While the latter is vector addition as defined above, the former is the addition operation defined on the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} .

8. Vector addition is distributive over scalar multiplication

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(\vec{u}+\vec{v})=a\vec{u}+a\vec{v}} . In this case vector addition takes place on both sides of the equality.

9. Scalar multiplication is associative

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a(b\vec{u})=(ab)\vec{u}} . This means that the algebraic structure of the underlying field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is preserved. Note that the left-hand side of this equality contains two subsequent applications of the scalar multiplication defined above, while the right-hand side contains one scalar multiplication as defined in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} (that of $ab$), followed by scalar multiplication with the vector ${\vec {u}}$.

10. The multiplicative identity of $F$ provides a scalar multiplicative identity

$1{\vec {u}}={\vec {u}}$, where $1$ is the multiplicative identity of the field $F$.

Properties 1 - 5 state that a vector space is an Abelian group with addition as group operation.

These axioms can be expressed concisely in mathematical notation as follows:

$\forall {\vec {u}},{\vec {v}},{\vec {w}}\in V,\ \forall a,b\in F,$

1. ${\vec {u}}+{\vec {v}}\in V$
2. ${\vec {u}}+{\vec {v}}={\vec {v}}+{\vec {u}}$
3. ${\vec {u}}+({\vec {v}}+{\vec {w}})=({\vec {u}}+{\vec {v}})+{\vec {w}}$
4. $\exists {\vec {0}}\in V:{\vec {0}}+{\vec {u}}={\vec {u}}+{\vec {0}}={\vec {u}}$
5. $\exists \ -\!{\vec {u}}\in V:-{\vec {u}}+{\vec {u}}={\vec {u}}+(-{\vec {u}})={\vec {0}}$
6. $a{\vec {u}}\in V$
7. $(a+b){\vec {u}}=a{\vec {u}}+b{\vec {u}}$
8. $a({\vec {u}}+{\vec {v}})=a{\vec {u}}+a{\vec {v}}$
9. $a(b{\vec {u}})=(ab){\vec {u}}$
10. $1{\vec {u}}={\vec {u}}$

Some important theorems

Linear dependence

A system of p ( ≥ 1 ) vectors ${\vec {u}}_{1},\dots ,{\vec {u}}_{p}$ of a vector space V is called linearly dependent if there exist coefficients (elements in F ) a₁, ..., a_p not all zero, such that the linear combination is the zero vector in V,

$\sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}\in V.$

Otherwise, the vectors ${\vec {u}}_{1},\dots ,{\vec {u}}_{p}$ are called linearly independent. A single vector not equal to the zero vector is obviously linearly independent.

If all a₁, ..., a_p are zero (in F ) then   $\sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}.$   If the set ${\vec {u}}_{1},\dots ,{\vec {u}}_{p}$ is linearly independent then the relation   $\sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}$ implies that all a₁, ..., a_p are zero. Hence a set of p vectors in V is linearly independent if

$\sum _{\nu }\,a_{\nu }\,{\vec {u}}_{\nu }={\vec {0}}\quad \Longleftrightarrow \quad a_{\nu }=0,\quad {\hbox{for}}\quad \nu =1,\ldots ,p.$

Every set of vectors containing the zero vector is linearly dependent.

A system of linearly independent vectors ${\vec {u}}_{1},\dots ,{\vec {u}}_{p}$ remains linearly independent if some vectors are omitted from the system. For, let a subset of the first q vectors ${\vec {u}}_{1},\dots ,{\vec {u}}_{q}$, with q < p, be linearly dependent then one or more coefficients not equal to zero can be found while the following is true

$a_{1}{\vec {u}}_{1}+\cdots +a_{q}{\vec {u}}_{q}={\vec {0}}.$

Add to the left- and right-hand side of this expression   ${\vec {0}}=0\cdot {\vec {u}}_{q+1}+\cdots +0\cdot {\vec {u}}_{p}$   and we get a contradiction.

Dimension

In general there are infinitely many linearly independent vectors in a vector space. When the maximum number of linearly independent vectors is finite, say n, the vector space is called of finite dimension n. Otherwise the space is called infinite-dimensional. If V′ is a linear subspace of the n-dimensional space V (all elements of V′ belong simultaneously to V ), and V′ contains a set B of m linearly independent vectors then m < n, because B belongs to the n-dimensional space V. It follows that m is finite and that all subspaces of finite-dimensional spaces are finite-dimensional. If m is the maximum number of linearly independent vectors in V′ then this subspace is of dimension m < n. For finite n it can be shown that V′ coincides with V (is an "improper" subspace) if and only if n = m.