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Linear Independence and Vector Bases

This document discusses linear independence and bases of vector spaces. It defines key concepts such as linear combinations, spanning sets, and linearly independent sets. A set is linearly independent if the only way for a linear combination of its vectors to equal the zero vector is if all scalars are zero. A basis is a linearly independent set that spans the entire vector space. The document provides several examples of bases, such as the standard basis for Rn and the basis for the vector space of polynomials of degree at most n. It also states theorems about removing dependent vectors from a spanning set and finding a basis within a spanning set.

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Jerico Arciaga
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287 views2 pages

Linear Independence and Vector Bases

This document discusses linear independence and bases of vector spaces. It defines key concepts such as linear combinations, spanning sets, and linearly independent sets. A set is linearly independent if the only way for a linear combination of its vectors to equal the zero vector is if all scalars are zero. A basis is a linearly independent set that spans the entire vector space. The document provides several examples of bases, such as the standard basis for Rn and the basis for the vector space of polynomials of degree at most n. It also states theorems about removing dependent vectors from a spanning set and finding a basis within a spanning set.

Uploaded by

Jerico Arciaga
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd

Math 114: Linear Algebra

Linear Independence of Vectors and Bases of a Vector Space


Recall: Let V be a vector space and S = {v1 , . . . , vk } V .
A vector v V is said to be a linear combination of v1 , . . . , vk if there are scalars a1 . . . , ak R such that

v = a1 v1 + . . . + ak vk
The linear span of S is the set Span(S) of all linear combinations of v1 , . . . , vk
If Span(S) = W , then W is a vector space and We say that S is a spanning set for W .
Let b Rm and A be an m k matrix. is in the span of the columns of A if the system Ax = b has a solution.
Linear Independence and Basis
The set S is linearly independent (you can also say v1 , . . . , vk are linearly independent vectors) if whenever
a1 v1 + . . . + ak vk = 0, it will imply that a1 = a2 = . . . = ak = 0. Otherwise, S is said to be linearly
dependent.
Theorem: S = {v1 , . . . , vk } is linearly dependent if and only if one of the vi s can be written as a linear
combination of the other vectors in S. In particular, If 0 S, then S is linearly dependent.
If S is a linearly independent set that spans V , then we say that S is a basis for V .
Example:
   
1 0
1. , is a basis for R2
0 1

1 0 0
2. 0 , 1 0 is a basis for R3
0 0 1

3. {e1 , . . . , en } is a basis for Rn called the standard basis for Rn


4. {1, t, t2 } is a basis for P2 .
5. {1, t, . . . , tn } is a basis for Pn called the standard basis for Pn
        
1 0 0 1 0 0 0 0
6. E11 = , E12 = , E21 = , E22 = is a basis for M22
0 0 0 0 1 0 0 1
7. {Eij | i = 1, . . . , m and j = 1, . . . n} is a basis for Mmn called the standard basis for Mmn , where
Eij is the m n matrix with 1 in the (i, j) entry and 0 everywhere else.
8. Is {1, t, 4 t} a basis for P1 ?

3 4 2
9. Is 0 , 1 , 1 a basis for R3 ?
6 7 5


0 2 6
10. Let S = 2 , 2 , 16 and H = Span(S). Is S a basis for H?
1 0 5


1 2
11. Is 2 , 7 a basis for R3 ?
3 9

Theorem: Let S = {v1 , . . . , vk } and H = Span(S).


1. If S is linearly dependent, say vi is a linear combination of v1 , . . . , vi1 , then the set S = S \ {vi } also
spans H.
2. If H 6= {0}, some subset of S is a basis for H.
Two Views of the Basis: Suppose S is a basis for H.
1. S is a minimal spanning set for H. That is, if S S, then Span(S) H.
2. S is a maximal linearly independent set in H. That is, if S S H, then S is linearly dependent.
Remarks
1. A vector space can have several different bases.

1
2. If B is a spanning set for V (not necessarily linearly independent), then there is a subset of B that is a
basis for V .
3. If B is a basis for V , then any element of V can be uniquely expressed as a linear combination of elements
of B.
4. If B is linearly independent but not a spanning set for V , then we can find a v V such that C = B {v}
is still linearly independent. Note: Span(B) Span(C) V

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