Linear Algebra Note 7 Eigenvectors & Eigenvalues (2021.12.22 ~ 2021.12.29) Singular Value Decomposition (SVD) Theorem $A_{m \times n} = U_{m \times m} \Sigma_{m \times n} V_{n \times n}$ where $U$ and $V$ are orthogonal matrix (orthogonal columns) $\implies U^{-1} = U^T, V^{-1} = V^T$ $\Sigma = \left[\begin{array}{c c c c c c c} \sigma_1 & & & & & & \ & \sigma_2 & & & & & \

Linear Algebra Note 6 Eigenvectors & Eigenvalues (2021.12.08 ~ 2021.12.17) Eigenvalue Decomposition (EVD) or Matrix Diagonalization Theorem Let $A$ be a $n \times n$ matrix. $A$ can be decomposed into $A=S \Lambda S^{-1}$ or $\Lambda = S^{-1}AS$ (diagonalization) where $\Lambda = \left[\begin{array}{c c c c} \lambda_1 & 0 & \cdots & 0 \ 0 & \lambda_2 & \cdots & 0 \

Linear Algebra Note 5 Determinants (2021.11.24) Properties 9. $\begin{vmatrix} AB \end{vmatrix} = \begin{vmatrix} A \end{vmatrix}\begin{vmatrix} B

Linear Algebra Note 4 Orthogonality (2021.11.10 ~ 2021.11.17) Example $A = \left[\begin{array}{c c} 1 & 0 \ 1 & 1 \ 1 & 2 \ \end{array}\right],\ \vec{b} = \left[\begin{array}{c} 6 \

Linear Algebra Note 3 Vector Spaces and Subspaces (2021.10.27~2021.11.3) $A_{m \times n},\ rank(A) = r$ $R$ 的型式 r, m, n 的關係 矩陣種類 解的數量 case 1

Linear Algebra Note 2 3. Vector Spaces and Subspaces (2021.10.13 ~ 2021.10.22) Define: $\mathbb{R}^n = {(a_1,\ a_2,\ \cdots,\ a_n) \mid a_i \in \mathbb{R}}$ (The space $\mathbb{R}^n$ consists of all vectors $\vec{V}$ with $n$ components) Example $\left[\begin{array}{c c} 1 & 2 \ \end{array}\right] \in \mathbb{R}^2,\ \left[\begin{array}{c}

Linear Algebra Note 1 1. Introduction to Vectors (2021.09.22) Define: A vector is a collection of numbers in $\mathbb{R}$ $\vec{V} = \left[\begin{array}{c} 1 \ 1 \ \end{array}\right],\ \vec{W} = \left[\begin{array}{c} 2 \ 3 \