# 5: Vector Spaces

The two key properties of vectors are that they can be added together and multiplied by scalars, so we make the following definition.

Definition

A \(\textit{vector space}\) \((V,+,.\, ,\mathbb{R})\) is a set \(V\) with two operations \(+\) and \(\cdot\) satisfying the following properties for all \(u, v \in V\) and \(c, d \in \mathbb{R}\):

**(Additive Closure)**\(u+v \in V\). \(\textit{Adding two vectors gives a vector.}\)**(Additive Commutativity)**\(u+v=v+u\). \(\textit{Order of addition doesn't matter.}\)**(Additive Associativity)**\((u+v)+w = u+(v+w)\). \(\textit{Order of adding many vectors doesn't matter.}\)**(Zero)**There is a special vector \(0_V \in V\) such that \(u+0_V = u\) for all \(u\) in \(V\).**(Additive Inverse)**For every \(u \in V\) there exists \(w \in V\) such that \(u+w=0_V\).**(Multiplicative Closure)**\(c\cdot v \in V\). \(\textit{Scalar times a vector is a vector.}\)**(Distributivity)**\((c+d) \cdot v= c\cdot v + d\cdot v\). \(\textit{Scalar multiplication distributes over addition of scalars.}\)**(Distributivity)**\(c\cdot (u+v)= c\cdot u + c\cdot v\). \(\textit{Scalar multiplication distributes over addition of vectors.}\)**(Associativity)**\((cd)\cdot v = c \cdot (d \cdot v)\).**(Unity)**\(1\cdot v = v\) for all \(v \in V\).

Remark

Rather than writing \((V,+,.\, ,\mathbb{R})\), we will often say "let \(V\) be a vector space over \(\mathbb{R}\)''. If it is obvious that the numbers used are real numbers, then "let \(V\) be a vector space'' suffices. Also, don't confuse the scalar product with the dot product. The scalar product is a function that takes as inputs a number and a vector and returns a vector as its output. This can be written:

\[\cdot \colon \mathbb{R}\times V \rightarrow V\, .\]

Similarly

\[ +:V\times V \rightarrow V\, . \]

On the other hand, the dot product takes two vectors and returns a number. Succinctly: \(\cdot \colon V\times V \rightarrow \Re\). Once the properties of a vector space have been verified, we'll just write scalar multiplication with juxtaposition \(cv=c\cdot v\), though, to avoid confusing the notation.

### Contributor

David Cherney, Tom Denton, and Andrew Waldron (UC Davis)