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Mathematics LibreTexts

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.


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 \(v \in V\).


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:

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


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

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.