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.
- Operations: addition, scalar multiplication
- Examples: $\mathbb{R}^n$, $\mathbb{C}^n$, and $C^0[0,1]$
Scalar field ($F$)
- Operations: addition, multiplication
- 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.
- addition: addition of linear transformations.
- Scalar matrices: scalar multiple of the identity matrix.
Null space: 零空间
Hermitian adjoint: Hermite伴随
Matrix decompositions:
- QR decomposition $\sim$ Gram–Schmidt orthogonalization process.
Note here matrix $R$ is upper-triangular
$$X=QR$$
$$\{ \vec{x}_1, \cdots, \vec{x}_n \} \Rightarrow \{ \vec{q}_1, \cdots, \vec{q}_n \}$$
🏷 Category=Algebra Category=Linear Algebra