Vector Spaces (向量空间)

A vector space is a non-empty set $V$ of vectors that satisfies the following properties for any vectors $\vec{u}$, $\vec{v}$, and $\vec{w}$ in the vector space, and any scalar $c$ and $d$:

1. $\vec{u} + \vec{v} \in V$
2. $\vec{u} + \vec{v} = \vec{v} + \vec{u}$
3. $(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$
4. There exists a zero vector $\vec{0}$ in $V$ such that $\vec{u} + \vec{0} = \vec{u}$
5. For each vector $\vec{u}$, there exists a vector $-\vec{u}$ in $V$ such that $\vec{u} + (-\vec{u}) = \vec{0}$
6. $c\vec{u} \in V$
7. $c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}$
8. $(c + d)\vec{u} = c\vec{u} + d\vec{u}$
9. $c(d\vec{u}) = (cd)\vec{u}$
10. $1\vec{u} = \vec{u}$

Subspaces (子空间)

A subspace is a subset of a vector space that is itself a vector space under the same operations of vector addition and scalar multiplication.

To determine whether a subset $W$ of a vector space $V$ is a subspace, we only need to check the following three conditions:

1. The zero vector of $V$ is in $W$.
2. If $\vec{u}$ and $\vec{v}$ are in $W$, then $\vec{u} + \vec{v}$ is in $W$.
3. If $\vec{u}$ is in $W$ and $c$ is a scalar, then $c\vec{u}$ is in $W$.

Null Space (零空间)

The null space of a matrix $A$, denoted as $Nul A$, is the set of all vectors $x$ such that $A\vec{x} = 0$.

The null space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^n$.

Column Space (列空间)

The column space of a matrix $A$, denoted as $Col A$, is the set of all linear combinations of the columns of $A$.

If $A$ is an $m \times n$ matrix $\begin{bmatrix} \vec{a}_1 & \vec{a}_2 & \cdots & \vec{a}_n \end{bmatrix}$, then

$$Col A = Span\{\vec{a}_1, \vec{a}_2, \ldots, \vec{a}_n\}$$

The column space of an $m \times n$ matrix $A$ is a subspace of $\mathbb{R}^m$.

Info

The column space of an $m \times n$ matrix $A$ is $\mathbb{R}^m$ if and only if the columns of $A$ span $\mathbb{R}^m$, which is equivalent to saying that the equation $A\vec{x} = \vec{b}$ has a solution for every $\vec{b}$ in $\mathbb{R}^m$.

Row Space (行空间)

Tips

$\begin{bmatrix} a \\ b \\ c \end{bmatrix}$ is the same as $(a, b, c)$, while $\begin{bmatrix} a & b & c \end{bmatrix}$ is NOT the same as $(a, b, c)$.

If $A$ is an $m \times n$ matrix, then each row of $A$ can be viewed as a vector in $\mathbb{R}^n$. The row space of $A$, denoted as $Row A$, is the set of all linear combinations of the rows of $A$.

It is obvious that $Col A^T = Row A$.

Copyright

License under:Attribution-NonCommercial-ShareAlike 4.0 International (CC-BY-NC-SA-4.0)