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8.1 Dot Products and Orthogonal Sets

Definition: Suppose that $u=(u_1,\ldots,u_n)$ and $v=(v_1,\ldots,v_n)$ are both vectors in $\mathbb{R}^n$. Then the dot product of $u$ and $v$ is $u\cdot v = u_1v_1+\ldots+u_nv_n$.

Theorem: Let $u,v,w$ be in $\mathbb{R}^n$. Then the dot product has the following properties:

• (Symmetry) $u\cdot v = v\cdot u$,

• (Linearity) $(cu+v)\cdot w = cu\cdot w + v\cdot w$,

• (Positive Definite) $u\cdot u\geq 0$ for all $u$, and $u\cdot u=0$ if and and only if $u=0$.

Definition: Let $x$ be a vector in $\mathbb{R}^n$, then the norm of $x$ is given by $\|x\|=\sqrt{x\cdot x}$. Note that $\|cx\|=|c|\|x\|$.

For two vectors $u$ and $v$, the distance between $u$ and $v$ is given by $\|u-v\|$.

Definition: Let $u$ and $v$ be vectors in $\mathbb{R}^n$ are orthogonal if $u\cdot v=0$.

Theorem: (Pythagorean Theorem) Suppose that $u$ and $v$ are in $\mathbb{R}^n$. Then $\|u+v\|^2=\|u\|^2+\|v\|^2$ if and only if $u\cdot v=0$.

Theorem: (Triangle Inequality) If $u$ and $v$ are in $\mathbb{R}^n$, then $\|u+v\|\leq \|u\|+\|v\|$.

Definition: Let $S$ be a subspace of $\mathbb{R}^n$. A vector $u$ is orthogonal to $S$ if it is orthogonal to every vector in $S$. The set of all vectors orthogonal to $S$ is called the orthogonal complement of $S$ and is denoted $S^\perp$.

The orthogonal complement to a subspace is also a subspace.

Theorem: Let $B=\{v_1,\ldots,v_n\}$ be a basis for a subspace $S$. Then $u\in S^\perp$ ($u$ is orthogonal to $S$) if and only if $u$ is orthogonal to each $v_i$.

Example: Let $s_1=(1,0,-1)$ and $s_2=(1,1,1)$ and $S$ be the span of $s_1$ and $s_2$. Is $u=(-1,1,1)\in S^\perp$? What is a basis for $S^\perp$?

Definition: A set of vectors $V$ in $\mathbb{R}^n$ form an orthogonal set the vectors are pairwise orthogonal. This means that if $v_i$ and $v_j$ are distinct vectors in $V$, then $v_i\cdot v_j=0$.

Example:

• Is the standard basis an orthogonal set?

• What's a basis that is not orthogonal?

Theorem: An orthogonal set of nonzero vectors is linearly independent.

Definition: A basis that is orthogonal as a set is called an orthogonal basis. A basis that is orthogonal as a set and is comprised of vectors of norm 1 is called an orthonormal basis.

Theorem: Let $S$ be a subspace with orthogonal basis $\{v_1,\ldots,v_k\}$. Then any vector $s\in S$ can be written as a linear combination $v=c_1v_1+\ldots+c_kv_k$ with $c_i=v_i\cdot s/\|v_i\|^2$.

Example: (THIS IS A BAD EXAMPLE. TURNS OUT NOT TO BE ORTHOGONAL.) Let $v_1=(-2,1,1), v_2=(1,-1,-3), v_3=(4,7,-1)$. Write $(3,-1,5)$ as a linear combination of $v_i$.

For finite dimensional spaces, we have that $(S^\perp)^\perp=S$. Use this to show that every subspace is the null space of some matrix.