线性代数:向量-矩阵

08 May 2017

The contents are from chapter 1 of Introduction to Linear Algebra, Book of Gilbert Strang, editon 4th.

Review of the key ideas.

1 Vectors & Linear Combinations

A vector \(v\) in 2-dimensional space has two components \(v_1\) and \(v_2\).
\(v + w = (v_1 + w_1, v_2 + w_2)\) and \(cv = (cv_1, cv_2)\) are found a component at a time.
A linear combination of three vectors \(u\) and \(v\) and \(w\) is \(cu + dv + ew\).
Take all linear combinations of \(u\), or \(u\) and \(v\), or \(u, v, w\). In 3-dimensions, those combinations typically fill a line, then a plane, and the whole space \(R^3\).

The dot product \(v \cdot w\) multiplies each component \(v_i\) by \(w_i\) and adds all \(v_iw_i\)
The length \(\parallel v \parallel\) of a vector is the square root of \(v \cdot v\).
\(u = \frac{v}{\parallel v \parallel}\) is a unit vector. Its length is 1.
The dot product is \(v \cdot w = 0\) when vectors \(v\) and \(w\) are perpendicular.
The cosine of \(\theta\) (the angle between any nonzero \(v\) and \(w\)) never exeeds 1: \(\cos\theta = \frac{v \cdot w}{\parallel v \parallel \parallel w \parallel}\), Schwarz inequality: \(\mid v \cdot w \mid \leq \parallel v \parallel \parallel w \parallel\) .
Triangle inequality: \(\parallel v + w \parallel \leq \parallel v \parallel + \parallel w \parallel\).

Matrix times vector: \(Ax=\) combination of the columns of A.
The solution to \(Ax=b\) is \(x=A^{-1}b\), when \(A\) is an invertible matrix.
The difference matrix \(A\) is inverted by the sum matrix \(S=A^{-1}\).
The cyclic matrix \(C\) has no inverse. Its three columns lie in the same plane. Those dependent columns add to the zero vector. \(Cx=0\) has many solutions.

矩阵最基础的向量. 书中一开始就介绍向量的基本知识, 从而引导出矩阵内容. 这里要明白:

矩阵乘以向量, 从向量角度来理解.
矩阵是否可逆? 观察矩阵内各列向量是否独立.将矩阵的可逆性转换到向量的独立与否.
- Difference matrix A : \(\begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix}\)
- Dot products with rows: \(Ax = \begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} (1, 0, 0) \cdot (x_1, x_2, x_3) \\ (-1, 1, 0) \cdot (x_1, x_2, x_3) \\ (0, -1, 1) \cdot (x_1, x_2, x_3) \end{bmatrix}\). 可逆其列向量互相独立.
- Cyclic Differences \(Cx=\begin{bmatrix} 1 & 0 & -1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_1 - x_3 \\ x_2 - x_1 \\ x_3 - x_2 \end{bmatrix} = b\) Cyclic difference matrix \(C\) 是不可逆的, 其列向量不独立. 其中一个列向量可以由另外两个列向量线性表示.
Independence and Dependence: 独立与不独立.
- \(u, v, w\) are independent. No combination except \(0u+0v+0w=0\) gives \(b=0\)
- \(u, v, w^{1}\) are dependent. Other combinations(specifically \(u + v + w^{1}\)) give \(b=0\).
- Note: 向量是否独立, 可以判断矩阵A的可逆性：
  - 矩阵A所有列向量互相独立, 那么矩阵A可逆(invertible matrix).
  - 矩阵A所有列向量不互相独立, 那么矩阵A不可逆(singular matrix).