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Linear independence, orthogonal basis and orthogonal matrix

2022-06-09 09:42:00 【Poor and poor to an annual salary of millions】

Preface

Before that 【 Understanding matrix series 】 Articles and 【 Understand eigenvalues and eigenvectors 】 All mentioned Linearly independent and The base The concept of , And in the follow-up study, the concept is confused or the definition is not understood clearly , Now let's sort it out systematically .
The content is my own learning summary , Many of them learn from others , Finally, a link is given .

Definition

1、 The definition of linear independence ： In linear algebra , A set of elements in a vector space , If there is no vector, it can be represented by a finite number of linear combinations of other vectors , It is called linear independent or linear independent , The other way around is called linear correlation .
For example, in three-dimensional Euclidean space R Three vectors of $(1, 0, 0), (0, 1, 0)$ and $(0, 0, 1)$ Linearly independent ; but $(2, - 1, 1), (1, 0, 1)$ and $(3, - 1, 2)$ Linear correlation , Because the third is the sum of the first two .
2、 The base ： The base (Basis) Is a set of linearly independent vectors ( aggregate ), Through their linear combination , Can be combined into any element in space (Span). That is, the basis is linearly independent .
3、 Orthogonal basis ： test $2$ Whether the vectors are orthogonal , It depends on whether their inner product is $0$ . Orthogonal basis is that each vector is orthogonal to each other .
Be careful ： Bases cannot be parallel , Intersection required , Orthogonality is not mandatory , But the orthogonality of Kita is different , The orthogonal basis of unit length is the most fragrant , Standard orthogonal basis .
Relationship ： A set of vectors that are linearly independent can be used as a set of bases , But this set of bases is not necessarily orthogonal , It must not be parallel .