A Swift and Brutal Introduction to Linnear Alfebra!

0 Graphics' Dependencies... Graphics' Dependencies

Basic Mathematics
- linear algebra linear algebra
- calculus
- Statistics statistics
basic physics
- Optical Optics
- Mechanics
other
- signal processing signal processing
- numerical analysis numerical analysis
- A bit of aesthetics

1 Vector Vectors

1.1 Vector Normalization of Vector Normalization

$\hat{a} = \vec{a}/|a|$

1.2 Vector Addition Vector Addition

Geometry Understanding Grometrically
- Parallelogram law & Triangle law
algebraically
- 各坐标相加 Simply add coordinates
  $\boldsymbol{A}=4x+3y,\; \boldsymbol{A}=\begin{pmatrix}x\\y\end{pmatrix},\; \boldsymbol{A}^T=(x,y),\; |\boldsymbol{A}|=\sqrt{x^2+y^2} \quad$

1.3 Vector dot product Dot(scalar) Product

1.3.1 Definition and usage

$\vec{a}\cdot\vec{b} =|\vec{a}||\vec{b}|\cos\theta,\; \cos\theta=\frac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|\cos\theta}$
$\quad cos\theta=\hat{a}\cdot\hat{b}$

1.3.2 Properties

$\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}$
$\vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot \vec{b}+\vec{a}\cdot\vec{c}$
$(k\vec{a})\cdot\vec{b}=\vec{a}\cdot(k\ vec{b})=k(\vec{a}\cdot\vec{b})$

1.3.3 Dot Product in Cartesian Coordinates

In 2D
$\vec{a}\cdot\vec{b}=\begin{pmatrix} x_a \\ y_a \\ \end{pmatrix}\cdot\begin{pmatrix} x_b \\ y_b \\ \end{pmatrix}=x_ax_b+y_ay_b$
In 3D
$\vec{a}\cdot\vec{b}\cdot\vec{c}=\begin{pmatrix} x_a \\ y_a \\ z_a \end{pmatrix}\cdot\begin{pmatrix} x_b \\ y_b \\ z_b\end{pmatrix}=x_ax_b+y_ay_b+z_az_b$

1.3.4 Dot Product for Graphics

Find angle between two vectors (eg cosine of angle between light source and surface)
Finding projection of one vector on another

$\vec{b}_\bot=k\hat{a},\;k=|\vec{b}_\ bot|=|\vec{b}|\cos\theta$
Measure how close two directions are
Decompose a vector Decompose a vector
Judgment towards Determine forward / backward

1.4 Cross product of vector Cross(vector) Product

1.4.1 Definition and usage

The result of the cross product is a vector, perpendicular to the two initial vectors Cross product is orthogonal to two initial vectors
The result of the cross product is determined by the right-hand rule Direction determined by right-hand rule
Useful in constructing coordinate systems
The modulus of the cross product result is equal in size to the area of the triangle formed by the two vectors
$|\vec{a}\times\vec{b}| =|\vec{a}||\vec{b}|\sin\theta$

1.4.2 Properties

$\vec{a}\times\vec{a}=\vec{0}$
$\vec{a}\times\vec{b}=-\vec{b}\times\vec{a}$
$\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times \vec{b}+\vec{a}\times\vec{c}$
$\vec{a}\times(k\vec{b})=k(\vec{a}\times\vec{b})$

1.4.3 Cross Product: Cartesian Formula in the Cartesian coordinate system

$\vec{a}\times\vec{b}=\begin{pmatrix} y_az_b-y_bz_a \\ z_ax_b-x_az_b \\x_ay_b-y_ax_b \end{pmatrix}$

1.4.4 Cross Product in Graphics

determine left and right
- As shown in the figure below, in the right-handed coordinate system, the result of cross multiplying vector b by a vector is positive, then b is on the left side of a
judging inside and outside
- As shown in the figure below, if the result of cross multiplying three times in order is positive or negative, it is internal, otherwise it is external

1.5 Orthonormal Bases / Coordinate Frames

$\quad|\vec{u}|=|\vec{v}|=|\vec{w}|=1,\;\vec{u}\cdot\vec{v}+\vec {v}\cdot\vec{w}+\vec{u}\cdot\vec{w}=0,\;\vec{w}=\vec{u}\times\vec{v}$
$刪\quad \vec{p}=(\vec{p}\cdot \vec{u})\vec{u}+(\vec{p}\cdot\vec{v})\vec{v}+(\vec{p}\cdot\vec{w})\vec{w }$

2 Matrix Matrix

In Graphics,
pervasively used to represent transformations

2.1 What is a matrix What is a matrix

Array of numbers (m × n = m rows, n columns)
Addition and multiplication by a scalar are trivial: element
by element

2.2 Matrix-Matrix Multiplication

The number of columns in A must equal to the number of rows in B
$(M\times N )(N\times P)=(M\times P)$
The element (i,j) in the product is the dot product result of row i in A and column j in B
$\ Begin {pmatrix} 1 & 3 \\ 5 & 2 \ \ \ 0 & 4 \ end {pmatrix} \ begin {pmatrix} 3 & 6 & 9 & 4 \ 8 & 8 & end {pmatrix} = \ be = \ be gin {pmatrix} 9 & 27 & 33 & 13 \\ 19 & 44 & 61 & 26 \\8&28&32&? \end{pmatrix}$
$?=(0,4)\cdot(4,3)=0+12=12$

2.3 Properties of the matrix

Non-commutative(AB and BA are different in general)
Associative and distributive
$(AB)C=A(BC),\;A(B+C)=AB+ AC$

2.4 Matrix-Vector Multiplication

Treat vector as a column matrix (m×1)
Is the core of the transformation Key for transforming points
$y-axis:\quad\begin{pmatrix} -1&0\\0&1\ \ \end{pmatrix}\begin{pmatrix} x \\ y \\ \end{pmatrix}=\begin{pmatrix} -x \\ y \\ \end{pmatrix}$

2.5 Transpose of a Matrix

Interchange the rows and columns of the matrix Switch rows and columns (ij -> ji)
$\begin{pmatrix} 1&2\\3&4\\5&6 \end {pmatrix}^T=\begin{pmatrix} 1&3&5\\2&4&6\\ \end{pmatrix}$
The nature of transposition Property
$AB)^T=B^TA^T$

2.6 Identity Matrix and Inverses

Identity Matrix
- The diagonals of the identity matrix are all 1
  $I_{3×3}=\begin{pmatrix} 1&0&0\\0&1&0\\0&0&1 \end{pmatrix}$
Inverse matrix Inverses
- The product of two matrices is an identity matrix, then the two matrices are inverse
  $AA^{-1}=A^{-1}A=I$
- Properties of Inverse Matrix
  $AB)^{-1}=B^{-1}A^{-1}$

3 Vector multiplication in Matrix form

3.1 Dot product

$\vec{a}\cdot\vec{b}=\vec{a}^T\vec{b}=\begin{pmatrix} x_a&y_a&z_a\end{pmatrix}\begin{pmatrix} x_b\\y_b\\z_b \end{pmatrix}=\begin{pmatrix} x_ax_b&y_ay_b&z_az_b\end{pmatrix}$

3.2 Cross product

$\vec{a}\times\vec{b}=A^ *\vec{b}=\underbrace{\begin{pmatrix} 0&-z_a&y_a\\z_a&0&-x_a\\-y_a&x_a&0 \end{pmatrix}}_{\rm{a vector adjoint matrix (dual\,matrix)} }\begin{pmatrix} x_b\\y_b\\z_b \end{pmatrix}$

GAMES101 Notes_Lec02_Review of Linear Algebraphics