Vector (向量)

Definition

A vector is an $m \times 1$ matrix (a special matrix that has only one column):

$$\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix}$$

The set of all $m$-dimensional vectors (vectors that have $m$ components) is denoted by $\R^m$ (read “r-m space”).

Specifically, we define $\vec{0}$ as the $m$-dimensional zero vector:

$$\vec{0} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$$

Calculation

The calculation of vectors is defined in the same way as the calculation of matrices.

$$\vec{u} + \vec{v} = \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_m \end{bmatrix} + \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix} = \begin{bmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_m + v_m \end{bmatrix}$$

$$c \vec{v} = c \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix} = \begin{bmatrix} cv_1 \\ cv_2 \\ \vdots \\ cv_m \end{bmatrix}$$

Matrix as a Collection of Column Vectors

A matrix can be viewed as a collection of column vectors:

$$A = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix} = \begin{bmatrix} | & | & & | \\ \vec{a_1} & \vec{a_2} & \cdots & \vec{a_n} \\ | & | & & | \end{bmatrix} \text{ , where } \vec{a_j} = \begin{bmatrix} a_{1,j} \\ a_{2,j} \\ \vdots \\ a_{m,j} \end{bmatrix}$$

Specifically, we define $\vec{e_1}, \vec{e_2}, \ldots, \vec{e_n}$ as the column vectors of $I_n$:

$$I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{bmatrix}_{n \times n} = \begin{bmatrix} | & | & & | \\ \vec{e_1} & \vec{e_2} & \cdots & \vec{e_n} \\ | & | & & | \end{bmatrix}$$

Linear Combinations and Span

A linear combination (线性组合) of vectors $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$ with scalars $c_1, c_2, \ldots, c_k \in \mathbb{R}$ is defined as:

$$c_1\vec{v_1} + c_2\vec{v_2} + \ldots + c_k\vec{v_k}$$

The span (张成空间) of $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$, or the subset of $\mathbb{R}^n$ spanned by $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$, is the collection of all linear combinations of these vectors, which can be expressed as:

$$Span\{\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}\} = \{c_1\vec{v_1} + c_2\vec{v_2} + \ldots + c_k\vec{v_k} \mid c_1, c_2, \ldots, c_k \in \mathbb{R}\}$$

Linear Independence (线性无关)

A set of vectors $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$ is said to be linearly independent (线性无关) if the linear equition $c_1\vec{v_1} + c_2\vec{v_2} + \ldots + c_k\vec{v_k} = \vec{0}$ has only the trivial solution $c_1 = c_2 = \ldots = c_k = 0$.

Otherwise, if there exists a nontrivial solution (at least one of $c_1, c_2, \ldots, c_k$ is not zero), then the vectors are said to be linearly dependent (线性相关).

According to the definition, we can conclude the following statements:

• If $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$ are linearly independent, then no vector in the set can be written as a linear combination of the other vectors.
• If a set of vectors contains $\vec{0}$, then the vectors are linearly dependent.
• If a set contains more vectors than there are entries in each vector, then the vectors are linearly dependent.

To determine whether a set of vectors $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$ is linearly independent, we can write the following matrix:

$$A = \begin{bmatrix} | & | & & | \\ \vec{v_1} & \vec{v_2} & \cdots & \vec{v_k} \\ | & | & & | \end{bmatrix}$$

The vectors $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$ are linearly independent if and only if $A$ has pivot positions in every column.

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