## Symbols

$\mathbb{R}^n, \mathbb{C}^n$: n-dementional vector spaces based on $\mathbb{R}$ and $\mathbb{C}$.

$\{ \vec{e}_1, \cdots, \vec{e}_n \}$: standard basis of $\mathbb{R}^n$ or $\mathbb{C}^n$.

$M_{m,n}$: set of all m-by-n matrices over $\mathbb{C}$.

$M_{n}$: set of all n-by-n matrices over $\mathbb{C}$.

## Concepts

Vector space ($V$): The fundamental setting for matrix theory.

• Examples: $\mathbb{R}^n$, $\mathbb{C}^n$, and $C^0[0,1]$

Scalar field ($F$)

• Examples: $\mathbb{R}$ and $\mathbb{C}$.

Subspace ($U$)

Span: 线性生成空间

Linear independence

Basis ($S$):

The intersection of class of linearly independent sets and class of span-to-space sets are bases.

• linearly independent set
• $\text{Span} S = V$

Dimension ($\text{dim}V$)

• Only for finite dimensional vector spaces

Isomorphism: vector spaces that can be related by invertible linear functions.

• For finite dimensional vector spaces $U,V$ over the same field, $$U \cong V \iff \dim U= \dim V$$
• For any basis $\mathcal{B}$ of an n-dimensional vector space $V$ over field $\mathbb{F}$, the mapping from a vector to its "coordinates" $f: \mathbf{x} \rightarrow [\mathbf{x}]_{\mathcal{B}}$ is an isomorphism between $V$ and $\mathbb{F}^n$.
• The above feature is analogous to random variables. It justified the re-abstraction (or second order abstration) from vector space $V$ to $F^n$, esp. $\mathbb{R}$ and $\mathbb{C}$.

Matrices

• Interpretations
• rectangular arrays of scalars
• linear transformation between two vector spaces based on $\mathbb{F}$.
• Operations
• multiplication: composition of linear transformations; associative.
• QR decomposition $\sim$ Gram–Schmidt orthogonalization process. Note here matrix $R$ is upper-triangular $$X=QR$$ $$\{ \vec{x}_1, \cdots, \vec{x}_n \} \Rightarrow \{ \vec{q}_1, \cdots, \vec{q}_n \}$$