MIT18.06 Linear Algebra (11)

lec11

matrix space

$S$ : space of 3x3 symmetric matrix : 6 dimension

$U$: space of 3x3 upper triangular matrix : 6 dimension

$S\cup U$ is not a matrix space.

$S\cap U $is a matrix space,$ \dim(S\cap U)=3$

$S+U $: any of$ S $**+** any of$ U$ = all 3x3’s.

$\dim(S+U)=9$

$\dim(S + U) + \dim(S \cap U) = \dim(S) + \dim(U) = 12$

solution to:

$$\frac{d^2y}{dx^2}+y=0$$

is

$$y=c_1\sin x+c_2\cos x$$

$y$ is a vector space, too.

All rank 1 matrix : $A=UVT$ where $U,V$ are column vectors.

All matrix can be a combination of rank one matrices.

All rank 4 matrices is not a matrix space. $rank(A + B) \le rank(A) + rank(B)$

$S $= all$ V $in$ \mathbb{R}^4 $with$ v_1+v_2 + v_3 + v_4 = 0$

$S $is a vector space,$ \dim(S) = 3$

$S $is null space to$ A = \begin{bmatrix}1 & 1 & 1 & 1\end{bmatrix} $i.e.$ AV = 0 $ $ S $是$ A = \begin{bmatrix}1 & 1 & 1 & 1\end{bmatrix} $的空空间，即$ AV = 0$ ,kms

$\dim N(A) = n − r$

e.g. $A = \begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}, r = 1, n = 4, \dim N(A) = 4 - 1 = 3$

$A = \begin{bmatrix}1 \text{(pivot)} & | & 1 & 1 & 1 \text{(free varibles)}\end{bmatrix}$

Basis for $S$ i.e. Special solution for $N(A)$ :

$\begin{bmatrix} -1\\1\\0\\0 \end{bmatrix}, \begin{bmatrix} -1\\0\\1\\0 \end{bmatrix}, \begin{bmatrix} -1\\0\\0\\1 \end{bmatrix}$

$N(AT)={0}, \dim N(AT) =0$

small world graph

Graph

Graph = {nodes, edges}