Open Mind

$\\ A = \begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix} , \indent B = \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix} , \indent C = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix} \\ \\ \\ D = \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}, \indent E = \begin{bmatrix} 1 & 0.2\\ 0 & 1 \end{bmatrix}, \indent F = \begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix},$

???

$R = \begin{bmatrix} cos\,\theta & -sin\, \theta \\ sin\, \theta & cos\, \theta \end{bmatrix}$

$\\ T \vec{e_1} = \begin{bmatrix} Cos \, \theta\\ Sin \, \theta \end{bmatrix} \\ \\ T \vec{e_2} = \begin{bmatrix} -Sin \, \theta \\ Cos \, \theta \end{bmatrix} \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix} = diagonal \, ?$

$\\ \sqrt{1^2 + 1^2} = \sqrt{2} \; \; \text{for some factor?} \; \sqrt{2} \; \; \text{from matrix} \\ \\ \sqrt{2} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\\ & \end{bmatrix} \\ \\ \\ \\ \begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix} = \sqrt{2} \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \\ \\ \indent \indent \text{with scalar ??}\\ \\ \\ = \sqrt{2} \begin{bmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4}\\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{bmatrix} \\ \\ \\ \eta ? = \begin{bmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{bmatrix}$

$\\ \begin{bmatrix} a & -b\\ b & a \end{bmatrix} \; \; \text{if} \; \; a^2 + b^2 = 1 \\ \\ \text {then without rotation scalar??} \\ \\ \text{column vector is unit vector}$

Matrix Projection

$PROJ_L\vec{x} = \vec{u} (\vec{u} \cdot \vec{x}) = \vec{u} ({\vec{u}\;^T\vec{x}}) =(\vec{u}\vec{u}\,^T)\vec{x}$

$\text{L is span} \; \left.\begin{matrix} s? \end{matrix}\right| \vec{u}$

$\vec{u} = \begin{bmatrix} u_{1}\\ u_{2} \end{bmatrix} \begin{bmatrix} u_{1} & u_{2} \end{bmatrix} \vec{x} = \underset{A}{\underbrace{\begin{bmatrix} {u_{1}}^2 & u_{2}u_{1}\\ u_{1}u_{2} & {u_{2}}^2 \end{bmatrix}}} \underset{\vec{x}}{\underbrace{\vec{x}}}$

$\\ \text{if L} = \text{span} (\vec{w}) \\ \\ ? \vec{u} = \frac{\vec{w}}{\left \| \vec{w} \right \|} \\ \\ \text{therefore matrix of Projection P} = \frac{1}{\vec{w}2}? \begin{bmatrix} {w_{1}}^2 & w_{1}w_{2}\\ w_{1}w_{2} & {w_{2}}^2 \end{bmatrix}$

Ch 2.2 Summary

$\text{Scaling} \indent \indent \indent \begin{bmatrix} 2 & 0\\ 0 & 2 \end{bmatrix}$

$\text{orthogonal projection onto horizontal axis} \indent \indent \indent \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}$

$\text{rotation? about y-axis (vertical)} \indent \indent \indent \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix}$

$\text{?? Rotation} \indent \indent \indent \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix}$

$\text{horizontal sheer} \indent \indent \indent \begin{bmatrix} 1 & 0.2\\ 0 & 1 \end{bmatrix}$

$\text{rotation combined with scaling} \indent \indent \indent \begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix}$

$\eta = \begin{bmatrix} cos \, \theta & -sin \, \theta\\ sin \, \theta & cos \, \theta \end{bmatrix} \indent \indent \indent \begin{bmatrix} a & -b\\ b & a \end{bmatrix}$

$\\ \text{if} \; a^2 + b^2 = 1 \; \text{then rotation without scaling} \\ \indent \indent \indent \indent \text{column vector is unit vector}$

$\text{ex} \; \begin{bmatrix} 1 & -1\\ 1 & 1 \end{bmatrix} = \sqrt{2} \begin{bmatrix} \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} = \sqrt{2} \begin{bmatrix} cos \, \frac{\pi}{\sqrt{4}} & - sin \, \frac{\pi}{\sqrt{4}}\\ sin \frac{\pi}{\sqrt{4}} & cos \frac{\pi}{\sqrt{4}} \end{bmatrix}$

$\\ \text{ex 2.2 problems 1-11 routine?} \\ \\ \text{sketch the ? of the standard L under the linear transformation or rotation: }$

$\text{\#1} \indent T\vec{x} = \begin{bmatrix} 3 & 1\\ 1 & 2 \end{bmatrix} \vec{x}$

Vector Visualization $\begin{matrix} \ \\ A = \\ \ \end{matrix}$ $\overset{\begin{bmatrix} 3 & 1\\ 1 & 2 \end{bmatrix}}{\rightarrow}$ Vector Visualization

$\\ T\begin{bmatrix} 1\\ 0 \end{bmatrix} = \begin{bmatrix} 3 & 1\\ 1 & 2 \end{bmatrix} \begin{bmatrix} 1\\ 0 \end{bmatrix} = \begin{bmatrix} 3\\ 1 \end{bmatrix} \\ \\ \\ T\begin{bmatrix} 0\\ 2 \end{bmatrix} = \begin{bmatrix} 3 & 1\\ 1 & 2 \end{bmatrix} \begin{bmatrix} 0\\ 2 \end{bmatrix} = \begin{bmatrix} 2\\ 4 \end{bmatrix}$

$\text{\#2} \indent \text{?? the rotation matrix through an angle 60}^{\circ} \; \text{in counter-clockwise direction}$

$R = \begin{bmatrix} cos \, \theta & -sin \, \theta \\ sin \, \theta & cos \, \theta \end{bmatrix} \indent \indent 60 ^{\circ} = \frac{\pi}{3}$

$R = \begin{bmatrix} cos \, \frac{\pi}{3} & -sin \, \frac{\pi}{3} \\ sin \, \frac{\pi}{3} & cos \, \frac{\pi}{3} \end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{bmatrix}$

$\\ \text{\#3} \indent \text{Consider a linear transformation} \; \; \text{T}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3} \\ \indent \indent \text{use} \; \text{T}(\vec{e_{1}}) \; \text{and} \; \text{T}(\vec{e_{2}}) \; \text{to describe the image of the unit square geometrically.}$

Vector Visualization $A = \begin{bmatrix} 1 & 0\\ 0 & 1\\ 0 & 0 \end{bmatrix}$ Vector Visualization

$\\ \text{\#4 Interpret } \\ \\ T\vec{x} = \begin{bmatrix} 1 & 1\\ -1 & 1 \end{bmatrix} x \\ \\ \text{Rotation ? by } \sqrt{2}$

$\\ T\vec{x} = \begin{bmatrix} 1 & 1\\ -1 & 1 \end{bmatrix} \vec{x} = \sqrt{2} \begin{bmatrix} cos \, \frac{-\pi}{4} & -sin\,(\frac{-\pi}{4})\\ sin \, \frac{-\pi}{4} & cos\,(\frac{-\pi}{4}) \end{bmatrix} \vec{x} \\ \\ \\ \text{T is a rotation through ?} (-\frac{\pi}{4})$

$\\ \text{\#5 The matrix} \\ \\ \indent \begin{bmatrix} -0.8 & -0.6\\ 0.6 & -0.8 \end{bmatrix} \\ \\ \text{represents a rotation. Find the ? of rotation(in radians)} \\ \\ R= \begin{bmatrix} cos \, \theta & -sin \, \theta\\ sin \, \theta & cos \, \theta \end{bmatrix} \\ \\ cos \, \theta = -0.8 \indent \indent \indent \indent \theta = cos^{-1}(-0.8)\times 2.498?$

$\text{\#6 Let be the with } \mathbb{R}^{3} \text{ that consists of all scalar multiples of vector } \begin{bmatrix} 2\\ .\\ 2 \end{bmatrix} , \; \text{ find the orthogonal projection of } \begin{bmatrix} .\\ .\\ . \end{bmatrix} \text{ on L}$

$PROJ_Lx = (\vec{u_{1}} \cdot x) \vec{u_{1}}$ $\vec{x} = \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}$

$PROJ_L \cdot \vec{x} = (\frac{\vec{x} \cdot \vec{w}}{w \cdot w}) \vec{w}$ $L = \text{span}(\begin{bmatrix} 2\\ ? \end{bmatrix})?$

$=(\frac{\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}\begin{bmatrix} 2\\ 1\\ 2 \end{bmatrix}}{\begin{bmatrix} 2\\ 1\\ 2 \end{bmatrix}\begin{bmatrix} 2\\ 1\\ 2 \end{bmatrix}}) (\begin{bmatrix} 2\\ 1\\ 2 \end{bmatrix})$

$=(\frac{5}{9})(\begin{bmatrix} 2\\ 1\\ 2 \end{bmatrix}) = \begin{bmatrix} \frac{10}{9}\\ \frac{5}{9}\\ \frac{10}{9} \end{bmatrix}$ $PROJ_{L} \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} = \begin{bmatrix} \frac{10}{9}\\ \frac{5}{9}\\ \frac{10}{9} \end{bmatrix}$

$\\ \text{\#7 Let } | \text{ be the line in } \mathbb{R}^{3} \text{ that consists of all scalar multiples of } \begin{bmatrix} 2\\ 1\\ 2 \end{bmatrix}. \text{ ? the reflection of the vector } \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \text{ about the line L} \\ \\ Ref =2 \, proj_{L} \, \vec{x} - \vec{x} \\ \\ proj_{L} \, \vec{x} = \begin{bmatrix} 10/9\\ 5/9\\ 10/9 \end{bmatrix} \\ \\ Ref = 2 \begin{bmatrix} 10/9\\ 5/9\\ 10/9 \end{bmatrix} - \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} \\ \\ = \begin{bmatrix} 20/9\\ 10/9\\ 20/9 \end{bmatrix} - \begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix} = \begin{bmatrix} 11/9\\ 1/9\\ 11/9 \end{bmatrix}$

$\\ \text{\#8 interpret geometrically} \\ \\ \text{T}(\vec{x}) = \begin{bmatrix} 0 & -1\\ -1 & 0 \end{bmatrix} \vec{x}$

E

H

I

L

P

R

W

Linear Algebra: Linear Transformations (II)

The Geometry of Linear Transformations

Matrix Projection

Ch 2.2 Summary