81 lines
1.9 KiB
TeX
Executable File
81 lines
1.9 KiB
TeX
Executable File
\section{Vectors}
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\definition{}
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Elements of $\mathbb{R}^n$ are often called \textit{vectors}. \\
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As you may already know, we have a few operations on vectors:
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\begin{itemize}
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\item Vector addition: $[a_1, a_2] + [b_1, b_2] = [a_1+b_1, a_2+b_2]$
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\item Scalar multiplication: $x \times [a_1, a_2] = [xa_1, xa_2]$.
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\end{itemize}
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\note{
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The above examples are for $\mathbb{R}^2$, and each vector thus has two components. \\
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These operations are similar for all other $n$.
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}
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\problem{}
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Compute the following or explain why you can't:
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\begin{itemize}
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\item $[1, 2, 3] - [1, 3, 4]$ \note{Subtraction works just like addition.}
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\item $4 \times [5, 2, 4]$
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\item $a + b$, where $a \in \mathbb{R}
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^5$ and $b \in \mathbb{R}^7$
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\end{itemize}
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\vfill
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\problem{}
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Consider $(2, -1)$ and $(3, 1)$ in $\mathbb{R}^2$. \\
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Can you develop geometric intuition for their sum and difference?
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\begin{center}
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\begin{tikzpicture}[scale=1]
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\draw[->]
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(0,0) coordinate (o) -- node[below left] {$(1, 2)$}
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(2, -1) coordinate (a)
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;
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\draw[->]
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(a) -- node[below right] {$(3, 1)$}
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(5, 0) coordinate (b)
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;
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\draw[
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draw = gray,
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text = gray,
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->
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]
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(o) -- node[above] {$??$}
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(b) coordinate (s)
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;
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\end{tikzpicture}
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\end{center}
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\vfill
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\pagebreak
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\definition{Euclidean Norm}
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In general, a \textit{norm} on $\mathbb{R}^n$ is a map from $\mathbb{R}^n$ to $\mathbb{R}^+_0$ \\
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Usually, one thinks of a norm as a \say{length metric} on a vector space. \\
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The norm of a vector $v$ is written $||v||$. \\
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\vspace{2mm}
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We usually use the \textit{euclidean norm} when we work in $\mathbb{R}^n$. \\
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If $v \in \mathbb{R}^n$, the euclidean norm is defined as follows:
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$$
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||v|| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}
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$$
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This is simply an application of the pythagorean theorem.
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\problem{}
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Compute the euclidean norm of
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\begin{itemize}
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\item $[2, 3]$
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\item $[-2, 1, -4, 2]$
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\end{itemize}
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\vfill
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\pagebreak |