Vector Space

We first define addition and multiplication on a set \(\setv\).

• Addition

  Addition on set \(\setv\) is a function that assigns an element \(u + v \in \setv\) for every element \(u, v \in \setv\).

• Scalar Multiplication

  Multiplication on set \(\setv\) is a function that assigns an element \(\lambda v \in \setv\) for every element \(\lambda \in \setv\) and \(v \in \setv\).

\(\field\) is a set of real or complex numbers.

A Vector Space is a set \(\setv\) with addition and multiplication defined on \(\setv\) and the following properties hold.

• Commutativity

  \(u + v = v + u\) where both \(u,v \in \setv\)

• Associativity

  \((u + v) + w = u + (v + w)\) and \((ab)v = a(bv)\) for all \(u,v,w \in \setv\) and \(a,b \in \field\)

• Additive Identity

  There exsits an element \(0 \in \setv\) such that \(v + 0 = v\) for all \(v \in \setv\). This element is unique. Mutilication of any element \(a \in \field\) by \(0\) results in \(0\).

• Additive Inverse

  For every element \(v \in \setv\) there exists \(w \in \setv\) such that \(v + w = 0\). This element is unique. Because of the uniqueness, the additive inverse of \(v\) is denoted by \(-v\) such that \(v + (-v) = 0\). Multiplying an element with the scalar \(-1\) also results in the additive inverse of that element.

• Multiplicative Identity There exists an element \(1 \in \field\) such that \(1 \times v = v\) for all \(v \in \setv\)

• Distributive Property

  \(a(u + v) = au + av\) and \((a + b)v = av + bv\) for all \(u,v \in \setv\) and \(a,b \in \setv\)

Elements of a vector space are called vectors or points. As we have noticed in the definitions above, the correct way to describe the vector space is to write vector space over a field \(\field\). A vector space over \(\real\) is called a real vector space and a vector space over \(\comp\) is called a complex vector space.