線性代數基本概念
阿新 • • 發佈：2020-11-07
線性代數概念的理解
Vector / Matrix
What’s a vector?
向量實際上是具有n 維屬性的一個較為複雜的客觀實體
\[x = \{x_1,x_2,...,x_n\} \]Linear transformation:the essence of Matrix
矩陣實際上代表一個線性變換
Basis
Basis
一組可以張成該空間的，線性無關的向量的集合稱為一組基
Dimension
The number of vectors in any basis of V is called the dimension of V, and is written dim(V).
Bases as Coordinate Systems
Span 張成的子空間
線性組合
\[S\left( {{\alpha _1},...,{\alpha _n}} \right) = \left\{ {\sum\limits_{i = 1}^n {{c_i}{\alpha _i}\;|\;{c_i} \in \mathbb{R}} } \right\} \]which is the set of all linear combinations of the vectors in this subspace.
Column Space 列空間
列向量的所有線性組合的集合構成的子空間
Row Space
Null Space (Kernel) 零空間 & Nullity
Intuative Theorems
\[\dim(\text{rowspace}(A)) = \dim(\text{colspace}(A)) = \text{rank}(A) \\ \text{rank}(A) + \text{nullity}(A) = n \]Rank
列空間的維數叫做秩
The number of dimensions in the column space.
Linearly Dependent 線性相關
at least one of the vectors in the set can be written as a