MATH 323 Lecture 19

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Norm

$\overrightarrow{x}\in {\mathbb{ℝ}}^n$

$\Vert \overrightarrow{x}\Vert =\sqrt{\sum _{i=1}^n{x_i}^2}$, where $\Vert \overrightarrow{x}\Vert$ is notation for the norm or length of $\overrightarrow{x}$.

$\Vert \overrightarrow{x}\Vert =\Vert -\overrightarrow{x}\Vert$

Distance between two points/vectors: $\Vert \overrightarrow{x}-\overrightarrow{y}\Vert$

${\Vert \overrightarrow{x}\Vert }^2=\overrightarrow{x}\cdot \overrightarrow{x}=\sum _{i=1}^n{x_i}^2$

Theorem 5.1.1

If $\overrightarrow{x},\overrightarrow{y}\in {\mathbb{ℝ}}^n$ and $\theta$ is the angle between them, then

$\overrightarrow{x}\cdot \overrightarrow{y}=\Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert \cos \theta$

Law of Cosines

Triangle defined by $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{x}-\overrightarrow{y}$

${\Vert \overrightarrow{y}-\overrightarrow{x}\Vert }^2={\Vert \overrightarrow{x}\Vert }^2+{\Vert \overrightarrow{y}\Vert }^2-2\Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert \cos \theta$

The above can be simplified to the form

$\overrightarrow{x}\cdot \overrightarrow{y}=\Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert \cos \theta$

Unit Vector

$\overrightarrow{u}$ is a unit vector iff $\Vert \overrightarrow{u}\Vert =1$

Unit vector in direction of $\overrightarrow{x}$ is given by

$\overrightarrow{u}=\hat{x}=\frac{\overrightarrow{x}}{\Vert \overrightarrow{x}\Vert }$

Example

$\begin{array}{rlrl}\overrightarrow{x}& =⟨3,4⟩& \overrightarrow{y}& =⟨-1,7⟩\\ \overrightarrow{u}& =\frac{1}{5}⟨3,4⟩=⟨\frac{3}{5},\frac{4}{5}⟩\\ \overrightarrow{v}& =\frac{1}{5\sqrt{2}}⟨-1,7⟩=⟨-\frac{1}{5\sqrt{2}},\frac{7}{5\sqrt{2}}⟩\\ \cos \theta & =\overrightarrow{u}\cdot \overrightarrow{v}=\dots =\frac{1}{\sqrt{2}}\\ \theta & =\frac{pi}{4}\end{array}$

Cauchy-Schwartz Inequality

$|\overrightarrow{x}\cdot \overrightarrow{y}|\le \Vert x\Vert \Vert y\Vert ⟺-1\le \frac{\overrightarrow{x}\cdot \overrightarrow{y}}{\Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert }\le 1$

Equality iff one of the vectors is zero or one of the vectors is a multiple of another ($\theta =0,\pi$)

Orthogonality

$\overrightarrow{x}$ and $\overrightarrow{y}$ are orthogonal (written $\overrightarrow{x}\perp \overrightarrow{y}$ if $\theta =\frac{\pi }{2},\frac{3\pi }{2}⟺\overrightarrow{x}\cdot \overrightarrow{y}=0$

Scalar and Vector Projections

Given two noncolinear vectors $\overrightarrow{x}$ and $\overrightarrow{y}$

Let $\overrightarrow{u}=\frac{\overrightarrow{y}}{\Vert \overrightarrow{y}\Vert }$ be a unit vector in the direction of $\overrightarrow{y}$ and $\overrightarrow{p}=\alpha \overrightarrow{u}$ be the vector projection of $\overrightarrow{x}$ in the direction of $\overrightarrow{y}$, where $\alpha$ is the scalar projection of $\overrightarrow{x}$ in the direction of $\overrightarrow{y}$

$\begin{array}{rl}\Vert \overrightarrow{p}\Vert =\alpha & =\Vert \overrightarrow{x}\Vert \cos \theta \\ & =\frac{\Vert \overrightarrow{x}\Vert \Vert \overrightarrow{y}\Vert \cos \theta }{\Vert \overrightarrow{y}\Vert }\\ & =\frac{\overrightarrow{x}\cdot \overrightarrow{y}}{\Vert \overrightarrow{y}\Vert }\end{array}$

Exercise

Given a point $P(1,4)$ and a line $y=\frac{1}{3}x$, find the point on the line closest to $P$.

Find vector $\overrightarrow{w}$ on the line: $\overrightarrow{w}=⟨3,1⟩$

Take vector projection of $\overrightarrow{v}=P-O$ onto $\overrightarrow{w}$:

$\overrightarrow{p}=\frac{\overrightarrow{v}\cdot \overrightarrow{w}}{\Vert \overrightarrow{w}\Vert }\cdot \overrightarrow{w}=⟨2.1,0.7⟩$

Planes in 3D Space

(See MATH 251 Lecture 5#Planes→)

Generalization of Pythagorean Theorem

Given two orthogonal vectors $\overrightarrow{x}$ and $\overrightarrow{y}$ in ${\mathbb{ℝ}}^n$,

$\begin{array}{rl}{\Vert \overrightarrow{x}\pm \overrightarrow{y}\Vert }^2& =(\overrightarrow{x}\pm \overrightarrow{y})\cdot (\overrightarrow{x}\pm \overrightarrow{y})\\ & =\overrightarrow{x}\cdot \overrightarrow{x}\pm 2\overrightarrow{y}\cdot \overrightarrow{x}+\overrightarrow{y}\cdot \overrightarrow{y}\\ & ={\Vert \overrightarrow{x}\Vert }^2+{\Vert \overrightarrow{y}\Vert }^2\end{array}$

Orthogonal Subspaces

Two subspaces $X$ and $Y$ in ${\mathbb{ℝ}}^n$ are said to be orthogonal if $\overrightarrow{x}\cdot \overrightarrow{y}=0$ for all $\overrightarrow{x}\in X$ and $\overrightarrow{y}\in Y$.

$A$ is $m\times n$ matrix

$N(A)$ is subspace, $\overrightarrow{x}\in {\mathbb{ℝ}}^n$, $A\overrightarrow{x}=0$ iff $a_{i1}{x}_1+a_{i2}{x}_2+\dots +a_{in}+x_n=0$ for $i=1,\dots ,m$.

This means that $\overrightarrow{x}$ is perpendicular to the $i$^th row of $A$

...

Orthogonal Complement

$Y\subset {\mathbb{ℝ}}^n$

Orthogonal complement given by

$Y^{\perp }=\{\overrightarrow{x}\in {\mathbb{ℝ}}^n\mid \overrightarrow{x}\cdot \overrightarrow{y}=0\mathrm{\forall }\overrightarrow{y}\in \overrightarrow{y}\}$

Example

Let $Y=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}({\overrightarrow{e}}_1)$ be the $x$-axis.

$Y^{\perp }=\mathrm{S}\mathrm{p}\mathrm{a}\mathrm{n}({\overrightarrow{e}}_2,{\overrightarrow{e}}_3)$ is the $yz$-plane

Theorem

1. If $X\perp Y$, then $X\cap Y=\{\overrightarrow{0}\}$
2. If $Y$ is a subspace of ${\mathbb{ℝ}}^n$, then $Y^{\perp }$ is also a subspace of ${\mathbb{ℝ}}^n$

Proof

1. Proof by contradiction. $X\cap Y\ne \{\overrightarrow{0}\}⟶(\overrightarrow{x}\in X\cap Y)\cdot (\overrightarrow{x}\in X\cap Y)={\Vert x\Vert }^2\ne 0$ — CONTRADICTION!
2. $\overrightarrow{u},\overrightarrow{v}\in Y^{\perp }$, so $\overrightarrow{u}\cdot \overrightarrow{y}=\overrightarrow{v}\cdot \overrightarrow{y}=0\mathrm{\forall }\overrightarrow{y}\in Y$. $(\alpha \overrightarrow{u}+\beta \overrightarrow{v})\cdot \overrightarrow{y}=0$, so subspace defined by $\alpha \overrightarrow{u}+\beta \overrightarrow{v}\in Y^{\perp }$

Fundamental Subspaces

$m\times n$ matrix $A$

$A\overrightarrow{x}=\overrightarrow{b}$ for some $\overrightarrow{x}\in {\mathbb{ℝ}}^n$ iff $\overrightarrow{b}$ is in col space of $A$.

We can write $A$ as a linear transformation $L_A:{\mathbb{ℝ}}^n\to {\mathbb{ℝ}}^n$.

The column space of $A$ is also called the range of $A$:

$R(A)=\{\overrightarrow{b}\in {\mathbb{ℝ}}^m\mid \overrightarrow{b}=A\overrightarrow{x}\mathrm{\exists }\overrightarrow{x}\in {\mathbb{ℝ}}^n\}$

Range of transpose matrix

$R(A^T)=\{\overrightarrow{y}\in {\mathbb{ℝ}}^n\mid \overrightarrow{y}=A^T\overrightarrow{x}\mathrm{\exists }\overrightarrow{x}\in {\mathbb{ℝ}}^m\}$

Thus $R(A)\subset {\mathbb{ℝ}}^m$, and $R(A^T)\subset {\mathbb{ℝ}}^n$, and

$R(A^T)\perp N(A)$

Theorem 5.2.1

Fundamental Subspace Theorem

$N(A)=R(A^T{)}^{\perp }$

$N(A^T)=R(A{)}^{\perp }$