MIT18.06 Linear Algebra (5)

lec5

$A=LU$

$\forall $invertible$ A $,$ \exists PA=LU$

transpose

$$\left(A^{\top}\right)_{i j}=A j i$$

Symmetrix Matrix

$$A^{\top}=A$$

$R^{\top}R$ is always symmetric.

proof. $\left(R^{\top} R\right)^{\top}=R^{\top} R$

Vector Space

e.g.

$\mathbb{R} ^2$ is all 2-d real vectors.

Vector space is closed to addition and multiplication.

Subspace

Subspace of $\mathbb{R}^2$ :

• $\mathbb{R}^2$
• any line through $\begin{bmatrix} 0\\0\end{bmatrix}$
• only $0$ vector

Subspace of matrices

All the combinations of the column vectors of $A$ is called column space $C(A)$ .