Matrix Theory study notes[6]

文章目录

• linear space
• references

linear space

1. a basis of linear space V k V^k Vk,which is x 1 , x 2 , . . . x k x_1,x_2,...x_k x1​,x2​,...xk​,can be called as a coordinate system.let vector v ∈ V k v \\in V^k v∈Vk and it can be linear expressed on this basis as v = a 1x 1 + a 2x 2 + . . . + a kx k v=a_1x_1+a_2x_2+...+a_kx_k v=a1​x1​+a2​x2​+...+ak​xk​,the a 1 , a 2 , . . . . , a k a_1,a_2,....,a_k a1​,a2​,....,ak​ is coordinate in this coordinate system denoted by ( a 1 , a 2 , . . . , a k) T (a_1,a_2,...,a_k)^T (a1​,a2​,...,ak​)T.
2. the various coordinate systems for the same vector are different usually because of non-uniqueness of basis of a linear space. for the first basis which is x 1 , x 2 , . . . x k x_1,x_2,...x_k x1​,x2​,...xk​ ,the coordinate is ( a 1 , a 2 , . . . , a k) T (a_1,a_2,...,a_k)^T (a1​,a2​,...,ak​)T and there are the second basis x 1 ′ , x 2 ′ , . . . x k ′ x_1\',x_2\',...x_k\' x1′​,x2′​,...xk′​ to coorespond another coordinate ( a 1 ′ , a 2 ′ , . . . , a k ′) T (a_1\',a_2\',...,a_k\')^T (a1′​,a2′​,...,ak′​)T,also can be explain that v = a 1x 1 + a 2x 2 + . . . + a kx k = a 1 ′x 1 ′ + a 2 ′x 2 ′ + . . . + a k ′x k ′ v=a_1x_1+a_2x_2+...+a_kx_k=a_1\'x_1\'+a_2\'x_2\'+...+a_k\'x_k\' v=a1​x1​+a2​x2​+...+ak​xk​=a1′​x1′​+a2′​x2′​+...+ak′​xk′​.
3. let v ∈ V k v \\in V^k v∈Vk and x 1 , x 2 , . . . x k x_1,x_2,...x_k x1​,x2​,...xk​ is a basis of linear space,then $v$ can uniquely be separated into the linear combination that v = a 1x 1 + a 2x 2 + . . . + a kx k v=a_1x_1+a_2x_2+...+a_kx_k v=a1​x1​+a2​x2​+...+ak​xk​.

references

1. deepseek
2. 《矩阵论》