Vector Space (向量空间)

Definition

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. $1\vec{u} = \vec{u}$

Subspace (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 $W$.

Generalized Vector Spaces (广义向量空间)

The concept of vector spaces can be extended to include objects other than just column vectors. For example, the set of all polynomial functions of degree less than or equal to $n$ forms a vector space under the operations of polynomial addition and scalar multiplication:

$$P_n = \{a_0 + a_1x + a_2x^2 + \cdots + a_nx^n \mid a_i \in \mathbb{R}\}$$

However, when the elements of a vector space are not column vectors, we cannot use the coefficient matrix or augmented matrix to determine linear independence (as what we mentioned in the 3rd section). Instead, we need to determine whether some $\vec{v}_j (j > 1)$ in the vector set $\{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\}$ is a linear combination of the preceding vectors $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_{j-1}$.

Spaces of Matrices

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 $\mathbb{R}^n$.

Null space is also called the kernel of a matrix (矩阵的核), denoted as $Ker A$.

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 $Col A^T = Row A$.

Basis (基)

Definition

Let $W$ be a subspace of a vector space $V$. A set of vectors $\{\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_k\}$ in $W$ is called a basis for $W$ if the following two conditions are satisfied:

1. The vectors are linearly independent.
2. The vectors span $W$.

Note

1. A basis is not unique. For example, the set $\{(1, 0), (0, 1)\}$ and the set $\{(1, 1), (1, -1)\}$ are both bases for $\mathbb{R}^2$.
2. The 2nd condition includes the requirement that the vectors in the basis must be in $W$, since the span of the vector set includes every vector in the set.

Specifically, the set $\{\vec{e}_1, \vec{e}_2, \ldots, \vec{e}_n\}$ is called the standard basis for $\mathbb{R}^n$.

Finding Basis for Column Space and Row Space

To find a basis for the column space of an $m \times n$ matrix $A$, we can perform row operations to reduce $A$ to its row echelon form.

• To find a basis for the column space of $A$, we take the columns of $A$ (the ORIGINAL MATRIX) that correspond to the pivot columns in the row echelon form of $A$.
• To find a basis for the row space of $A$, we take the nonzero rows of the row echelon form of $A$ (the REDUCED MATRIX).

Tips

That is because: Row operations do not change a matrix's row space because each operation is applied to entire rows. However, row operations can change the column space.

However, do NOT take rows of the original matrix $A$ to find a basis for the row space of $A$, because row operations can change the order of the rows.

Dimension (维数)

Definition

The dimension of $V$, written as $dim V$, is the number of vectors in a basis for $V$.

If a vector space $V$ is spanned by a finite set, then $V$ is said to be finite-dimensional (有限维的); otherwise, $V$ is infinite-dimensional (无限维的).

Rank-Nullity Theorem (秩-零化度定理)

The rank of an $m \times n$ matrix $A$ (矩阵的秩), denoted as $rank A$, is the dimension of the column space of $A$.

The nullity of an $m \times n$ matrix $A$ (矩阵的零化度), denoted as $nullity A$, is the dimension of the null space of $A$.

The Rank-Nullity Theorem states that for any $m \times n$ matrix $A$,

$$rank A + nullity A = n$$

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