Linear Algebra lecture9 note

Linear independence

Spanning a space

Basis and dimension

以上概念都是针对a bunch of vectors, 不是矩阵里的概念

 

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Suppose A is m by n with m<n, then there are non-zero solutions to AX=0（more unknowns than equations)

Reason: There will be free variables

Independence:

Vectors X1, X2,…,Xn are independent if no combination gives zero vector( except the zero combination)

C1X1+C2X2+…+CnXn≠0

1.若以上向量中存在零向量，则不可能线性无关

2.平面内三个向量定成线性相关

3.如果零空间存在非零向量，那么各列线性相关

Repeat: when V1,V2,…,Vn are columns of A,

they are independent if N(A) is only zero vectors( no free variable,r=n)

they are dependent if AC=0 for some non-zero C( has free variable,r<n)

 

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Spanning a space: Vectors V1,V2,..,Vl span a subspace means: The space consists of all combinations of those vectors

Basis: For a space is a sequence of vectors V1,V2,…,Vd with 2 properties:

1.They are independent

2.They span the spaces

Example:

space in R3

one space is

如何检验是否构成基？

可当作矩阵列向量，经过消元、变换，看是否能得到自由变量？是否列都是主列？

Rn，n vectors give basis if the n *n matrix with those columns if invertible

Given a space: Every basis for space has the same number of vectors, and this number is called dimension of space

 

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Summary:

Independence, that looks at combinations not being zero

(线性无关，着眼于线性组合不为0）

Spanning, that looks at all the combinations

(生成，着眼于所有的线性组合）

Basis, that’s the one that combines independence and spanning

(基，一组无关的向量并生成空间）

Dimension，the number of vectors in any basis

(维数，表示基向量的个数）