lec11 matrix space SSS : space of 3x3 symmetric matrix : 6 dimension UUU: space of 3x3 upper triangular matrix : 6 dimension S∪US\cup US∪U is not a matrix space. $S\cap U isamatrixspace,is a matrix sp

lec12 Graph Graph : Node, Edge nodes:n=4edges:m=5nodes : n = 4 \\ edges : m = 5nodes:n=4edges:m=5 e.g. An electrical network. Incidence Matrix (关联矩阵) A=[−11000−110−1010−100100−11]A = \begin{bmatrix}

lec14 orthogonal vectors（正交） x⊤y=0x^\top y=0x⊤y=0 ∥x∥2+∥y∥2=∥x+y∥2\| x\| ^2+\| y\| ^2=\| x+y\| ^2∥x∥2+∥y∥2=∥x+y∥2 Subspace Subspace S is orthogonal to subspace T means: every vector in S is orthogonal

lec16 Least square find a line y=C+Dxy=C+Dxy=C+Dx through: (1,1)(2,2)(3,2)(1,1)(2,2)(3,2)(1,1)(2,2)(3,2) : [111213][CD]=[122]\begin{bmatrix} 1 & 1\\ 1 & 2\\ 1 & 3\end{bmatrix}\begin{bmatri

lec17 Orthonormal vectors qi⊤qj={0 if i≠j1 if i=jq_{i}^\top q_j=\left\{\begin{array}{ll}0 & \text { if } i \neq j \\1 & \text { if } i=j\end{array}\right. qi⊤​qj​={01​ if i=j if i=j​ ⁍⁍ ⁍ So

lec18 ∣abcd∣=ad−bc\left|\begin{array}{ll}a & b \\c & d\end{array}\right|=a d-b c ​ac​bd​​=ad−bc determinate det⁡I=1\det I=1detI=1 Exchange rows, reverse sign of det⁡\detdet , so det⁡P=1\det P

lec19 2x2 det⁡\detdet ∣abcd∣=∣a0cd∣+∣0bcd∣=∣a0c0∣+∣a00d∣+∣0bc0∣+∣0b0d∣=ad−bc\left|\begin{array}{ll}a & b \\c & d\end{array}\right|=\left|\begin{array}{ll}a & 0 \\c & d\end{array}\right

lec20 2x2 inverse [abcd]−1=1ad−bc[d−b−ca]\left[\begin{array}{ll}a & b \\c & d\end{array}\right]^{-1}=\frac{1}{a d-b c}\left[\begin{array}{cc}d & -b \\-c & a\end{array}\right] [ac​bd​]−

lec7 null space the xxx such that Ax=[0000]Ax = \begin{bmatrix} 0\\0\\0\\0 \end{bmatrix}Ax=​0000​​always gives a vector space. Proof. ⁍⁍ ⁍ where CCC is a constant. rank rank of AAA = #pivots = #linea

lec8 Solve Ax=bAx=bAx=b [122224683680][x1x2x3x4]=[b1b2b3](Ax=B)\left[\begin{array}{llll}1 & 2 & 2 & 2 \\2 & 4 & 6 & 8 \\3 & 6 & 8 & 0\end{array}\right]\left[\begin{