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# 5: Vector Spaces

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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 $$c, d \in \mathbb{R}$$:

1. (Additive Closure) $$u+v \in V$$. $$\textit{Adding two vectors gives a vector.}$$
2. (Additive Commutativity) $$u+v=v+u$$. $$\textit{Order of addition doesn't matter.}$$
3. (Additive Associativity) $$(u+v)+w = u+(v+w)$$. $$\textit{Order of adding many vectors doesn't matter.}$$
4. (Zero) There is a special vector $$0_V \in V$$ such that $$u+0_V = u$$ for all $$u$$ in $$V$$.
5. (Additive Inverse) For every $$u \in V$$ there exists $$w \in V$$ such that $$u+w=0_V$$.
6. (Multiplicative Closure) $$c\cdot v \in V$$. $$\textit{Scalar times a vector is a vector.}$$
7. (Distributivity) $$(c+d) \cdot v= c\cdot v + d\cdot v$$. $$\textit{Scalar multiplication distributes over addition of scalars.}$$
8. (Distributivity) $$c\cdot (u+v)= c\cdot u + c\cdot v$$. $$\textit{Scalar multiplication distributes over addition of vectors.}$$
9. (Associativity) $$(cd)\cdot v = c \cdot (d \cdot v)$$.
10. (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.

• 5.1: Examples of Vector Spaces
One can find many interesting vector spaces, such as the following:
• 5.2: Other Fields
Above, we defined vector spaces over the real numbers. One can actually define vector spaces over any field. This is referred to as choosing a different base field. A field is a collection of "numbers'' satisfying certain properties.
• 5.3: Review Problems