update

src/master/COMP90014.md

@@ -78,4 +78,12 @@ tag:
   - scale-free
   - small-world
   - random
-  - regular
+  - regular
+
+## 4.2. Advanced Indexing
+
+- trie (prefix tree or radix tree)
+  - used in word retrieval
+- suffix tree
+  - complexity, space
+- suffix array

src/master/COMP90018.md

@@ -107,12 +107,13 @@ We work for an industry leading user interaction company. A medical institution
   - overshooting 超调
 
 
-###
+### Assignment 1 Discussion 2
 
 - who is the client
   - what current situation, to what motivation for change
   - regional hospital
   - CHI
   - ubitious computing
 - who can benefit
-  - patient
+  - patient
+

src/master/COMP90084.md

@@ -67,6 +67,48 @@ tag:
 
 ## Prequisites 2. Vector Spaces
 
+- addition: $(V+W)[j] = V[j] + W[j]$
+  - let $V = \begin{bmatrix} v_1 \\ v_2 \end{bmatrix}, W = \begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$
+  - then $V + W = \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix}$
+  - commutative: $V + W = W + V$
+  - associative: $(V + W) + X = V + (W + X)$
+- inverse: $V + (-V) = 0$
+- Abelian group 阿贝尔群: addition, inverse, associative, commutative
+- scalar: an abitrarily complex number
+- multiply an element by a scalar: $(c \cdot V)[j] = c \times V[j]$
+- transpose 转置: $V^T[j, k] = V[k, j]$
+- conjugate 共轭，复数相反，实数不变: $\overline{V}[j] = \overline{V[j]}$
+- adjoint/dagger 伴随，转置+共轭: $V^\dagger[j, k] = \overline{V[k, j]}$
+  - $(V^\dagger)^\dagger = V$
+  - $(c \cdot V)^\dagger = \overline{c} \cdot V^\dagger$
+- matrix multiplication: $\star$
+  - $(A \star B)^T = B^T \star A^T$
+  - $\overline{A \star B} = \overline{A} \star \overline{B}$
+  - $(A \star B)^\dagger = B^\dagger \star A^\dagger$
+- isomorphism 同构，双射（双向一一对应）
+  - 可以认为两个空间性质相同，仅命名不同
+- linear independence 线性无关
+  - linear combination: $V = c_1v_1 + c_2v_2 + \cdots + c_nv_n$
+  - 一组向量的线性组合不等于0
+  - $0=c_1v_1 + c_2v_2 + \cdots + c_nv_n$ only when $c_1 = c_2 = \cdots = c_n = 0$
+  - basis: a set of linearly independent vectors that span the space ${v_1, v_2, \cdots, v_n}$
+  - canonical/standard basis: a set of vectors with one 1 and the rest are 0
+  - dimension: the number of vectors in the basis
+- trace: sum of the diagonal elements
+  - $Tr(A) = \sum_{i=1}^{n} A[i, i]$
+- eigenvalue and eigenvector 特征值和特征向量
+  - for a matrix $A$, if $Av = \lambda v$, then $\lambda$ is the eigenvalue and $v$ is the eigenvector
+- symmetric matrix 对称矩阵
+  - $A = A^T$
+- hermitian matrix 共轭转置矩阵
+  - $A = A^\dagger$
+- invertible matrix 可逆矩阵
+  - $A^{-1} \star A = A \star A^{-1} = I$
+- unitary matrix 幺正矩阵
+  - $A^\dagger \star A = A \star A^\dagger = I$
+- tensor product 张量积
+  - $A \otimes B = \begin{bmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{bmatrix}$
+
 - orthogonal: 
   - for two vectors $v_1 = (a_1, b_1), v_2 = (a_2, b_2)$
   - if $v_1 \cdot v_2 = 0$, then $v_1$ and $v_2$ are orthogonal
@@ -109,6 +151,7 @@ tag:
 ## 1. Linear Algebra
 
 - inner product: two vertics give a number
+  - also called dot/scalar product
 - outer product: two vertics give a matrix
 
 ## 2. Quantum Systems