Lecture_002_Linear_Algbra_and_Vector_Calculus

Before preceded, you need to understand:

1. Vectors, Vector Space, and Euclidean Vector Space
2. Functions and see Functions (and Linear Transformations) as Vectors in Vector Space
3. Dot Product, Projections, Cross Product
4. Linear Transformation as matrix
5. Span, basis, linear independence, Gram-Schmidt Algorithm (QR decomposition)
6. Vector Field and 3D calculus

Vector

Norm: a non-negative quantity of elements of a vector space

• $\|\overrightarrow{u}\| \geq 0$

• $\|\overrightarrow{u}\| \iff u = \overrightarrow{0}$

• $\|c\overrightarrow{u}\| = \|c\| \cdot \|\overrightarrow{u}\|$

• $\|\overrightarrow{u}\| + \|\overrightarrow{v}\| \geq \|\overrightarrow{u} + \overrightarrow{v}\|$

Euclidean: length preserved by rotation, translation, reflection of space. $\|u\| := \sqrt{u_1^2 + ... + u_n^2}$ only if the vector is in orthonormal basis.

$L^2$ norm: magnitude of a function. $\|f\| := \sqrt{\int_a^b f(x)^2 dx}$ for $[a, b]$ is the function's domain

$L^2$ dot (inner) product: how well two functions "line up". $\langle f, g \rangle := \int_a^b f(x)g(x) dx$

Linear Maps (Transformation)

• Computational Cheap

• Capture Geometric Transformation (rotation, translation, scaling)

• All maps can be approximated as linear maps using Taylor's Series

Derivative and Integrals are Linear: Notes on Linear Transformations

Orthonormal Basis

Orthonormal Basis: $e_1 \cdot e_2 = \langle{e_i, e_j}\rangle = \begin{cases}1 \text{ if } i = j\\ 0 \text{ otherwise}\\\end{cases}$

1. unit length
2. mutually orthogonal

Fourier Transform

Approximated Signal: can be expressed using fourier basis of sinusoids.

Fourier Analysis (Decomposition): the process of transforming the original function (signal) into fourier series.

Fourier Composition: compose fourier series into approximated signal.

// QUESTION: do we use Discrete Time Fouier Series instead? // QUESTION: is Fourier Decomposition the same as Fourier Analysis?

Cross Product

Definition: $\sqrt{\det(u, v, u \times v)} = \|u\|\|v\|\sin \theta$, $u \times v := \begin{bmatrix} u_2v_3 - u_3v_2\\ u_3v_1 - u_1v_3\\ u_1v_2 - u_2v_1\\ \end{bmatrix}$

• In 2D: $u \times v := u_1v_2 - u_2v_1$

Lagrange's Identity: $u \times (v \times w) = v(u \cdot w) - w(u \cdot v)$

Derivatives

Gradient: $\triangledown f(x) \cdot u = D_u f(x)$

Example 1: taking partial derivative $\frac{\partial}{\partial u_k}$ of $f := u^T v$ for $1 \leq k \leq n$

\begin{align*} &\frac{\partial}{\partial u_k} u^Tv\\ &= \frac{\partial}{\partial u_k} \sum_{i = 1}^n u_iv_i\\ &= \sum_{i = 1}^n \frac{\partial}{\partial u_k}u_iv_i\\ &= \sum_{i = 1}^n \begin{cases} 0 \text{ if } i \neq k\\ v_k \text{ otherwise}\\ \end{cases}\\ &= v_k\\ & \implies \triangledown_u (u^Tv) = \begin{bmatrix} v_1\\ ...\\ v_n\\ \end{bmatrix} = v \end{align*}

Example 2: taking gradient of functions that takes functions as input: $F(f) := \langle f, g \rangle := \int_a^b f(x)g(x) dx$. We get $\triangledown F = g$

Divergence: $\triangledown \cdot X := \sum_{i = 1}^n \frac{\partial X_i}{\partial u_i}$ where $\triangledown = (\frac{\partial}{\partial u_1}, ..., \frac{\partial}{\partial u_n}), X(u) = (X_1(u), ..., X_n(u))$

Curl: $\triangledown \times X := \begin{bmatrix} \frac{\partial X_3}{\partial u_2} - \frac{\partial X_2}{\partial u_3}\\ \frac{\partial X_1}{\partial u_3} - \frac{\partial X_3}{\partial u_1}\\ \frac{\partial X_2}{\partial u_1} - \frac{\partial X_1}{\partial u_2}\\ \end{bmatrix}$ where $\triangledown = (\frac{\partial}{\partial u_1}, \frac{\partial}{\partial u_2}, \frac{\partial}{\partial u_3}), X(u) = (X_1(u), X_2(u), X_3(u))$

Laplacian

Laplacian: Operator used to encode concavity (concave up) for multivariable equation. Used in

• Fourier transform, frequency decomposition

• used to define model in partial differential equations (Laplace, heat, wave equation)

• characteristics of geometry

\begin{align*} \triangle f &:= \triangledown \cdot \triangledown f = \text{div}(\triangledown f) \tag{divergence of gradient}\\ \triangle f &:= \sum_{i = 1}^n \frac{\partial^2 f}{\partial x_i^2} \tag{sum of 2nd partial derivative}\\ \triangle f &:= - \triangledown_f(\frac{1}{2} \|\triangledown f\|^2) \tag{gradient of Dirichlet energy}\\ \triangle f &:= &\tag{graph Laplacian}\\ \triangle f &:= &\tag{variation of surface area}\\ \triangle f &:= &\tag{trace of Hessian}\\ \end{align*}

Hessian

\begin{align*} (\triangledown^2 f) u &:= D_u (\triangledown f)\\ \triangledown^2 f &:= \begin{bmatrix} \frac{\partial^2 f}{\partial x_1 \partial x_1} & ... & \frac{\partial^2 f}{\partial x_1 \partial x_n}\\ ... & ... & ...\\ \frac{\partial^2 f}{\partial x_n \partial x_1} & ... & \frac{\partial^2 f}{\partial x_n \partial x_n}\\ \end{bmatrix} \end{align*}

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