Vectors

A $n \times 1$ matrix is called a $n$ dimensional column vector or simply a $n$D vector:

$$\vec{w} = \begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix}$$

Sometimes, we will just call as vector if it is clear that is 2D.

$\mathbb{R}^n$ is the set of all $n$D vectors (向量空间) with real number entries (read r-N).

Specifically, we denote $\vec{0}$ as the zero vector (零向量):

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

Operations on Vectors

Vector Addition

Given two vectors $\vec{u}, \vec{v} \in \mathbb{R}^n$, their sum $\vec{u} + \vec{v}$ is defined as:

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

Scalar Multiplication

Given a vector $\vec{u} \in \mathbb{R}^n$ and a scalar $c \in \mathbb{R}$, the scalar multiplication $c\vec{u}$ is defined as:

$$c\vec{u} = \begin{bmatrix} cu_1 \\ cu_2 \\ \vdots \\ cu_n \end{bmatrix}$$

Linear Combinations

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

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

To determine whether a vector $\vec{b} \in \mathbb{R}^n$ can be expressed as a linear combination of vectors $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k} \in \mathbb{R}^n$, we can set up the following equation:

$$x_1\vec{v_1} + x_2\vec{v_2} + \ldots + x_k\vec{v_k} = \vec{b}$$

This equation can be rewritten in augmented matrix form:

$$\begin{bmatrix} v_{1,1} & v_{2,1} & \ldots & v_{k,1} & b_1 \\ v_{1,2} & v_{2,2} & \ldots & v_{k,2} & b_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ v_{1,n} & v_{2,n} & \ldots & v_{k,n} & b_n \end{bmatrix}$$

For brevity, write this matrix in a way that identifies its columns:

$$\begin{bmatrix} \vec{v_1} & \vec{v_2} & \ldots & \vec{v_k} & \vec{b} \\ \end{bmatrix}$$

Span (张成空间)

If $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k} \in \mathbb{R}^n$, then the set of all linear combinations of these vectors is denoted by

$$Span\{\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}\}$$

and is called the subset of $\mathbb{R}^n$ spanned by $\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}$.

In other words,

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

$\vec{b} \in Span\{\vec{v_1}, \vec{v_2}, \ldots, \vec{v_k}\}$ if and only if the augmented matrix $\begin{bmatrix} \vec{v_1} & \vec{v_2} & \ldots & \vec{v_k} & \vec{b} \\ \end{bmatrix}$ is consistent.

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