100 lines
3.1 KiB
TeX
100 lines
3.1 KiB
TeX
\section{Spaces}
|
|
|
|
\definition{Vector Spaces}
|
|
A \textit{space} over a field $\mathbb{F}$ consists of the following elements:
|
|
\begin{itemize}[itemsep = 2mm]
|
|
\item A set $V$, the elements of which are called \textit{vectors}
|
|
\item An operation called \textit{vector addition}, denoted $+$ \\
|
|
Vector addition operates on two elements of $V$. \\
|
|
|
|
\item An operation called \textit{scalar multiplication}, denoted $\times$ \\
|
|
Scalar multiplication multiplies an element of $V$ by an element of $\mathbb{F}$. \\
|
|
Any element of $\mathbb{F}$ is called a \textit{scalar}.
|
|
\end{itemize}
|
|
|
|
\vspace{2mm}
|
|
|
|
\textbf{Note:}
|
|
The same symbols are used for additions and multiplications in both $\mathbb{F}$ and $V$. \\
|
|
\textit{These are different operations}, so be aware of the context of each $+$ and $\times$.
|
|
|
|
\vspace{5mm}
|
|
|
|
Vector addition and multiplication must have the following properties. \\
|
|
In both tables, $x, y, z \in V$ and $a, b\in \mathbb{F}$.
|
|
|
|
\vspace{2mm}
|
|
|
|
% [t] and \vspace{0pt} ensure alignment at top
|
|
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
|
|
\begin{center}
|
|
\begin{tabular}{l | r@{=}l }
|
|
\hline
|
|
\multicolumn{3}{|c|}{Properties of vector addition} \\
|
|
\hline
|
|
Closure & \multicolumn{2}{c}{$x+y \in V$} \\
|
|
Associativity & $(x+y)+z~$&$~x+y+z$ \\
|
|
Commutativity & $x+y~$&$~y+x$ \\
|
|
Distributivity & $x(y+z)~$&$~xy + xz$ \\
|
|
Identity & $x+0~$&$~x$ \\
|
|
Inverse & $x + (-x)~$&$~0$
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{minipage}%
|
|
\hfill%
|
|
\begin{minipage}[t]{0.48\textwidth}\vspace{0pt}
|
|
\begin{center}
|
|
\begin{tabular}{l | r@{=}l }
|
|
\hline
|
|
\multicolumn{3}{|c|}{Properties of vector multiplication} \\
|
|
\hline
|
|
Closure & \multicolumn{2}{c}{$ax \in V$} \\
|
|
Distributivity & $a(x+y)~$&$~ax+ay$ \\
|
|
& $(a+b)x~$&$~ax+bx$ \\
|
|
Compatibility$^*$ & $(ab)x~$&$~a(bx)$ \\
|
|
Identity & $a+0~$&$~a$
|
|
\end{tabular}
|
|
\end{center}
|
|
\end{minipage}
|
|
|
|
\vspace{2mm}
|
|
|
|
$^*$ Remember that $a, b \in \mathbb{F}$ and $x \in V$. Thus, $(ab)$ is scalar multiplication and $(bx)$ is vector multiplication. \\
|
|
Compatibility is \textit{not} associativity, it is \say{compatibility of vector and scalar multiplication.}
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
\definition{}
|
|
There is a good chance you are familiar with basic vector arithmetic. \\
|
|
Here's a quick review:
|
|
\begin{itemize}
|
|
\item Scalar multiplication is done elementwise: $3 \times [a, b, c] = [3a, 3b, 3c]$.
|
|
\item Vector addition is similar: $[a, b, c] + [1, 2, 3] = [a+1,~b+2,~c+3]$.
|
|
\item Vector addition is not valid for vectors of different sizes.
|
|
\end{itemize}
|
|
|
|
|
|
\definition{}
|
|
We usually use the \textit{dot product} as our vector product. It is defined as follows. \\
|
|
Given two vectors $a, b \in \mathbb{R}^n$, the dot product $a \cdot b$ is $\sum_1^n a_ib_i$.
|
|
|
|
\vspace{2mm}
|
|
|
|
In other words, if $a = [1, 2, 3]$ and $b = [4, 5, 6]$,
|
|
$$
|
|
a \cdot b = (1 \times 4) + (2 \times 5) + (3 \times 6) = 32
|
|
$$
|
|
As you may expect, the dot product $ab$ is valid iff $a$ and $b$ are the same size.
|
|
|
|
|
|
|
|
\problem{}
|
|
Show that the dot product satisfies the properties of a vector product listed above. \\
|
|
Conclude that $\mathbb{R}^n$ is a vector space over $\mathbb{R}$.
|
|
|
|
\vfill
|
|
\pagebreak
|
|
|
|
|