Linear Combination, Sets and Spans

This is easy Steven, you know.

This also covers the Basis. Spanning Set

Show that two spans are equal

To show that $\text{Span A}=\text{Span B}$, we show that $\text{Span A}\subseteq \text{Span B}$ and $\text{Span B}\subseteq \text{Span A}$.

K-Flats and Hyperplanes

Definition 17.2: For some positive integer $k\ge n-1$, let $\overrightarrow{v_1},\dots ,\overrightarrow{v_k}\in {\mathbb{R}}^n$ be such that $\{\overrightarrow{v_1},\dots ,\overrightarrow{v_k}\}$ is linearly independent. The set with vector equation $\vec{x}=\vec{p}+c_1\overrightarrow{v_1}+\cdots +c_k\overrightarrow{v_k}{c}_1,\dots ,c_k\in \mathbb{R}$

is a k-flat in ${\mathbb{R}}^n$ through the point $P$.

Definition 17.3: Let $\overrightarrow{v_1},\dots ,{\vec{v}}_{n-1}\in {\mathbb{R}}^n$ be such that $\{\overrightarrow{v_1},\dots ,{\vec{v}}_{n-1}\}$ is linearly independent. The set with vector equation

$\vec{x}=\vec{p}+c_1\overrightarrow{v_1}+\cdots +c_{n-1}{\vec{v}}_{n-1}{c}_1,\dots ,c_{n-1}\in \mathbb{R}$

is a hyperplane in ${\mathbb{R}}^n$ through the point $P$.

Orthogonal Sets and Bases

A set $\{\overrightarrow{v_1},\dots ,\overrightarrow{v_k}\}\subseteq {\mathbb{R}}^n$ is an orthogonal set if $\overrightarrow{v_i}\cdot \overrightarrow{v_j}=0$ for $i\ne j$

If $\vec{x}\in \text{span}\mathbb{S}$, then $\vec{x}=\frac{\vec{x}\cdot \overrightarrow{v_1}}{\Vert \overrightarrow{v_1}{\Vert }^2}\overrightarrow{v_1}+\cdots +\frac{\vec{x}\cdot \overrightarrow{v_k}}{\Vert \overrightarrow{v_k}{\Vert }^2}\overrightarrow{v_k}$

Orthonormal Sets and Bases

An orthogonal set $\{v_1,\dots ,v_k\}\subseteq {\mathbb{R}}_n$ is called an orthonormal set if $||\overrightarrow{v_i}||=1$ for $i=1,\dots ,k$. If an orthonormal set $B$ is a basis for subspace $S$ of ${\mathbb{R}}_n$, then $B$ is an orthonormal basis for $S$.