3.7: Supplements - Vector Space
Introduction
You have long taken for granted the fact that the set of real numbers, \(\mathbb{R}\), is closed under addition and multiplication, that each number has a unique additive inverse, and that the commutative, associative, and distributive laws were right as rain. The set \(\mathbb{C}\), of complex numbers also enjoys each of these properties, as do the sets \(\mathbb{R}^{n}\) and \(\mathbb{C}^n\) of columns of n real and complex numbers, respectively.
To be more precise, we write \(\textbf{x}\) and \(\textbf{y}\) in \(\mathbb{R}^{n}\) as
\(\textbf{x} = (x_{1}, x_{2}, \cdots, x_{n})^{T}\)
\(\textbf{y} = (y_{1}, y_{2}, \cdots, y_{n})^{T}\)
and define their vector sum as the elementwise sum
\[\textbf{x}+\textbf{y} = \begin{pmatrix} {x_{1}+y_{1}}\\ {x_{2}+y_{2}}\\ {\vdots}\\ {x_{n}+y_{n}} \end{pmatrix} \nonumber\]
and similarly, the product of a complex scalar, \(\textbf{z} \in \mathbb{C}\) with \(\textbf{x}\) as:
\[\textbf{zx} = \begin{pmatrix} {zx_{1}}\\ {zx_{2}}\\ {\vdots}\\ {zx_{n}} \end{pmatrix} \nonumber\]
Vector Space
These notions lead naturally to the concept of vector space. A set \(V\) is said to be a vector space if
- \(\textbf{x}+\textbf{y}=\textbf{y}+\textbf{x}\) for each \(\textbf{x}\) and \(\textbf{y}\) in \(V\).
- \(\textbf{x}+\textbf{y}+\textbf{z} = \textbf{y}+\textbf{x}+\textbf{z}\) for each \(\textbf{x}\), \(\textbf{y}\) and \(\textbf{z}\) in \(V\).
- There is a unique "zero vector" such that \(\textbf{x}+\textbf{0} = \textbf{x}\) for each \(\textbf{x}\) in \(V\).
- For each \(\textbf{x}\) in \(V\) there is a unique vector \(-\textbf{x}\) such that \(\textbf{x}+ -\textbf{x} = \textbf{0}\).
- \(1 \textbf{x} = \textbf{x}\).
- \((c_{1}c_{2}) \textbf{x} = c_{1}(c_{2} \textbf{x})\) for each \(\textbf{x}\) in \(V\) and \(c_{1}\) and \(c_{2}\) in \(\mathbb{C}\).
- \(c(\textbf{x}+\textbf{y}) = c\textbf{x}+c\textbf{y}\) for each \(\textbf{x}\) and \(\textbf{y}\) in \(V\) and c in \(\mathbb{C}\).
- \((c_{1}+c_{2}) \textbf{x} = c_{1} \textbf{x}+c_{2} \textbf{x}\) for each \(\textbf{x}\) in \(V\) and \(c_{1}\) and \(c_{2}\) in \(\mathbb{C}\).