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Linear algebra-Basis & Dimension
1. LINEAR ALGEBRA
BASIS AND DIMENSION
MANIKANTA SATYALA
Department of Mathematics
VSM COLLEGE(A), Ramachandrapuram
2. Definition : Basis
A basis of a vector space V is an ordered set of linearly
independent (non-zero) vectors that spans V.
Notation: 1 , , nฮฒ ฮฒ
Definition :- Basis
A subset S of a vector space V(F) is said to be the basis of V, if
i) S is linearly independent
ii) The linear span of S is V i.e., L(S)=V
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
3. Example :
2 1
,
4 1
B
๏ฆ ๏ถ ๏ฆ ๏ถ
๏ฝ ๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ
is a basis for R2
B is L.I. :
2 1 0
4 1 0
a b
๏ฆ ๏ถ ๏ฆ ๏ถ ๏ฆ ๏ถ
๏ซ ๏ฝ๏ง ๏ท ๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ ๏จ ๏ธ
โ
2 0
4 0
a b
a b
๏ซ ๏ฝ
๏ซ ๏ฝ
โ
0
0
a
b
๏ฝ
๏ฝ
B spans R2:
2 1
4 1
x
a b
y
๏ฆ ๏ถ ๏ฆ ๏ถ ๏ฆ ๏ถ
๏ฝ ๏ซ๏ง ๏ท ๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ ๏จ ๏ธ
โ
2
4
a b x
a b y
๏ซ ๏ฝ
๏ซ ๏ฝ
โ
๏จ ๏ฉ
1
2
2
a y x
b x y
๏ฝ ๏ญ
๏ฝ ๏ญ
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
4. Example :
1 2
,
1 4
B
๏ฆ ๏ถ ๏ฆ ๏ถ
๏ข ๏ฝ ๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ
is a basis for R2 that differs from B only in order.
Definition : Standard / Natural Basis for Rn
1 0 0
0 1 0
, , ,
0 0 1
n
๏ฆ ๏ถ ๏ฆ ๏ถ ๏ฆ ๏ถ
๏ง ๏ท ๏ง ๏ท ๏ง ๏ท
๏ง ๏ท ๏ง ๏ท ๏ง ๏ท๏ฝ
๏ง ๏ท ๏ง ๏ท ๏ง ๏ท
๏ง ๏ท ๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ ๏จ ๏ธ
1 2, , , n๏ฝ e e e
๏จ ๏ฉi ikk
๏ค๏ฝekth component of ei =
1
0
i k
for
i k
๏ฝ๏ฌ
๏ฝ ๏ญ
๏น๏ฎ
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
6. Example:
For the function space
๏ป ๏ฝcos sin ,a b a b๏ฑ ๏ฑ๏ซ ๏
a natural basis is cos , sin๏ฑ ๏ฑ
Another basis is cos sin , 2cos 3sin๏ฑ ๏ฑ ๏ฑ ๏ฑ๏ญ ๏ซ
Proof is straightforward.
Example :
For the function space of cubic polynomials P3 ,
a natural basis is 2 3
1, , ,x x x
Other choices can be
3 2
, 3 , 6 , 6x x x
2 2 3
1,1 ,1 ,1x x x x x x๏ซ ๏ซ ๏ซ ๏ซ ๏ซ ๏ซ
Proof is again straightforward.
Rule: Set of L.C.โs of a L.I. set is L.I. if each L.C. contains a different vector.
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
7. Example : Matrices
Find a basis for this subspace of M2๏ด2 : 2 0
0
a b
a b c
c
๏ฌ ๏ผ๏ฆ ๏ถ๏ฏ ๏ฏ
๏ฝ ๏ซ ๏ญ ๏ฝ๏ญ ๏ฝ๏ง ๏ท
๏จ ๏ธ๏ฏ ๏ฏ๏ฎ ๏พ
2
,
0
b c b
b c
c
๏ฌ ๏ผ๏ญ ๏ซ๏ฆ ๏ถ๏ฏ ๏ฏ
๏ฝ ๏๏ญ ๏ฝ๏ง ๏ท
๏จ ๏ธ๏ฏ ๏ฏ๏ฎ ๏พ
Solution:
1 1 2 0
,
0 0 1 0
b c b c
๏ฌ ๏ผ๏ญ๏ฆ ๏ถ ๏ฆ ๏ถ๏ฏ ๏ฏ
๏ฝ ๏ซ ๏๏ญ ๏ฝ๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ๏ฏ ๏ฏ๏ฎ ๏พ
โด Basis is
1 1 2 0
,
0 0 1 0
๏ญ๏ฆ ๏ถ ๏ฆ ๏ถ
๏ง ๏ท ๏ง ๏ท
๏จ ๏ธ ๏จ ๏ธ
( Proof of L.I. is
left as exercise )
Theorem :
In any vector space, a subset is a basis if and only if each vector in the space can be
expressed as a linear combination of elements of the subset in a unique way.
Let i i i i
i i
c d๏ฝ ๏ฝ๏ฅ ๏ฅ ฮฒv ฮฒ then ๏จ ๏ฉi i i
i
c d๏ญ ๏ฝ๏ฅ ฮฒ 0
โด L.I. ๏ uniqueness
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
8. ๐. ๐ ๐ข๐ง๐ข๐ญ๐ ๐๐ข๐ฆ๐๐ง๐ญ๐ข๐จ๐ง๐๐ฅ ๐๐๐๐ญ๐จ๐ซ ๐ฌ๐ฉ๐๐๐
Definition :
A vector space V(F) is said to be finite dimensional if it
has a finite basis
or
A vector space V(F) is said to be finite dimensional if
there is a finite subset S in V such that L(S)=V
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
9. Theorem :-
๐ผ๐ ๐ ๐น ๐๐ ๐ ๐๐๐๐๐ก๐ ๐๐๐๐๐๐ก๐๐๐๐๐ ๐ฃ๐๐๐ก๐๐ ๐ ๐๐๐๐, ๐กโ๐๐ ๐กโ๐๐๐ ๐๐ฅ๐๐ ๐ก๐ ๐ ๐๐๐ ๐๐ ๐ ๐๐ก ๐๐ ๐
Proof :- Since V F is finite dimentional vector space
By the definition of finite dimentional vector space
there exists a finite set S such that L S = V
Let S = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑn
Assume that S does not contains 0 vector
If S is L.I., then S is a Basis set of V.
If S is L.D., then there exists a vector ๐ผ๐ โ S which can be expressed as
linear combination of its preceding vectors
Omitting vector ๐ผ๐ from S
Let S1 = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑiโ1, ฮฑi+1, โฆ , ฮฑn โ S1 โ S
๐ต๐ฆ ๐๐๐๐ค๐ ๐กโ๐๐๐๐๐ L S1 = L(S)
๐๐๐ค ๐ฟ ๐ = ๐ โ ๐ฟ ๐1 = ๐
๐ผ๐ ๐1 ๐๐ ๐ฟ. ๐ผ. ๐ ๐๐ก ๐กโ๐๐ ๐1 ๐ค๐๐๐ ๐๐ ๐ ๐๐๐ ๐๐ ๐๐ ๐.
๐ผ๐ ๐1 ๐๐ ๐๐ ๐ฟ. ๐ท. ๐กโ๐๐ ๐๐๐๐๐๐๐๐๐๐ ๐๐ ๐๐๐๐ฃ๐ ๐๐๐ ๐ ๐๐๐๐๐ก๐ ๐๐ข๐๐๐๐ ๐๐ ๐ ๐ก๐๐๐ ,
๐ค๐ ๐ค๐๐๐ ๐๐ ๐๐๐๐ก ๐ค๐๐กโ ๐ ๐ฟ. ๐ผ. ๐ ๐๐ก ๐ ๐ ๐๐๐ ๐ฟ ๐ ๐ = ๐. ๐ป๐๐๐๐ ๐ ๐ ๐ค๐๐๐ ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐๐ ๐
MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
10. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
Theorem :-
๐ฟ๐๐ก ๐ ๐น ๐๐ ๐ ๐๐๐๐๐ก๐ ๐๐๐๐๐๐ก๐๐๐๐๐ ๐ฃ๐๐๐ก๐๐ ๐ ๐๐๐๐ ๐๐๐ S = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑm
๐ ๐๐๐๐๐๐๐๐ฆ ๐๐๐๐๐๐๐๐๐๐๐ก ๐ ๐ข๐๐ ๐๐ก ๐๐ ๐. ๐โ๐๐ ๐๐๐กโ๐๐ ๐ ๐๐ก๐ ๐๐๐ ๐ ๐๐๐ ๐๐ ๐๐ ๐ ๐๐ ๐ ๐๐๐ ๐๐
๐๐ฅ๐ก๐๐๐๐๐ ๐ก๐ ๐๐๐๐ ๐ ๐๐๐ ๐๐ ๐๐ ๐
Proof :-
Since V F is finite dimentional vector space, it has a finite basis let it be B
Given S = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑm a linearly independent subset of V.
Let B = ฮฒ1, ฮฒ2, ฮฒ3, โฆ , ฮฒn
Now consider the set S1 = S โช B = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑm, ฮฒ1, ฮฒ2, ฮฒ3, โฆ , ฮฒn
๐๐๐๐๐๐๐ฆ ๐ฟ ๐ต = ๐
๐ธ๐๐โ ๐ผ ๐๐๐ ๐๐ ๐๐ฅ๐๐๐๐ ๐ ๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐ ๐ฝโฒ ๐ ๐๐ ๐ต ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐๐ ๐
โ ๐1 ๐๐ ๐ฟ. ๐ท.
๐ป๐๐๐๐ ๐ ๐๐๐ ๐ฃ๐๐๐ก๐๐ ๐๐ ๐1 ๐๐๐ ๐๐ ๐๐ฅ๐๐๐๐ ๐ ๐๐ ๐๐ ๐๐๐๐๐๐ ๐๐๐๐๐๐๐๐ก๐๐๐ ๐๐
๐๐ก๐ ๐๐๐๐๐๐๐๐๐๐ ๐ฃ๐๐๐ก๐๐.
๐โ๐๐ ๐ฃ๐๐๐ก๐๐ ๐๐๐๐๐๐ก ๐๐ ๐๐๐ฆ ๐๐ ๐ผโฒ ๐ , ๐ ๐๐๐๐ ๐ ๐๐ ๐ฟ. ๐ผ. ๐ ๐ ๐กโ๐๐ ๐ฃ๐๐๐ก๐๐ ๐๐ข๐ ๐ก ๐๐ ๐ ๐๐๐ ๐ฝ๐
L S1 = L S โช B = V ๐๐ L S โช B = ๐ฟ ๐ โช ๐ฟ ๐ต = ๐ฟ ๐ โช ๐ = ๐
12. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
๐๐จ๐ญ๐: โ๐
Every basis is a spanning set but converse need not true
If S is Basis of V then L S = V
but If L S = V for S โ V โ S
๐๐จ๐ญ๐: โ2
Let S = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑn be a basis set of a finite dimesional vector space V F
Then for every ฮฑ โ V there exists a unique set of scalars ๐ ๐, ๐ ๐, ๐ ๐, โฆ , ๐ ๐ง โ ๐
such that
๐ = ๐ ๐ ๐ ๐ + ๐ ๐ ๐ ๐ + ๐ ๐ ๐ ๐ + โฏ + ๐ ๐ง ๐ ๐ง
๐ = ๐ ๐ ๐ ๐ + ๐ ๐ ๐ ๐ + ๐ ๐ ๐ ๐ + โฏ + ๐ ๐ง ๐ ๐ง
If there exists other set of scalars ๐ ๐, ๐ ๐, ๐ ๐, โฆ , ๐ ๐ง โ ๐ such that
then ๐ ๐= ๐ ๐, ๐ ๐ = ๐ ๐, ๐ ๐= ๐ ๐,โฆ, ๐ ๐ง= ๐ ๐ง
13. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
3. COORDINATES
Definition : Coordinates
Let S = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑn be a basis set of a finite dimesional vector space V F .
Let ฮฒ โ V be given by
๐ = ๐ ๐ ๐ ๐ + ๐ ๐ ๐ ๐ + ๐ ๐ ๐ ๐ + โฏ + ๐๐ข ๐๐ข + โฏ + ๐ ๐ง ๐ ๐ง
for ๐ ๐, ๐ ๐, ๐ ๐, โฆ , ๐ ๐ง โ ๐
then the set ๐ ๐, ๐ ๐, ๐ ๐, โฆ , ๐ ๐ง are called the coordinates
15. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
๐ 2 โ ๐ 2 โ ๐ 1
1 1 5
0 1 โ2
0 1 โ2
๐ 3 โ ๐ 3 โ ๐ 2
1 1 5
0 1 โ2
0 0 0
Since there are only 2 non zero rows and 3 unknowns
Hence the given vectors are L.D.
Therefore given vectors donโt form basis
16. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
๐๐จ๐ญ๐: โ๐
Given set of vectors are L. I. if
1. In the coefficient matrix
๐๐ ๐๐ ๐๐๐๐๐๐ค๐๐ = ๐ ๐๐๐ ๐๐ ๐๐๐ก๐๐๐ฅ
Rank of Matrix = No of non โ zero rows
2. In the systerm of equations all coefficients are zeros
๐ + ๐ + 5๐ = 0
๐ + 2๐ + 3๐ = 0
2๐ + 5๐ + 4๐ = 0
โ ๐ = ๐ = ๐ = 0
Given set of vectors are L. D. if
1. In the coefficient matrix
๐๐ ๐๐ ๐๐๐๐๐๐ค๐๐ โ ๐ ๐๐๐ ๐๐ ๐๐๐ก๐๐๐ฅ
2. In the systerm of equations all coefficients are zeros
๐ + ๐ + 5๐ = 0
๐ + 2๐ + 3๐ = 0
2๐ + 5๐ + 4๐ = 0
โ ๐, ๐, ๐ ๐๐๐ก ๐๐๐ ๐ง๐๐๐๐
20. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
4. DIMENSION OF A VECTOR SPACE
Definition : DIMENSION OF A VECTOR SPACE
Let V F be the finite dimensional vector space. The Number of elements
in any basis of V is called the dimension of V and denoted by dim V
๐ฟ๐๐ก ๐ = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑ10 ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐๐ ๐(๐น)
For Example :-
๐ท๐๐๐๐๐ ๐๐๐ ๐๐ ๐ ๐น = dim ๐ = ๐ ๐ = ๐๐. ๐๐ ๐๐๐๐๐๐๐ก๐ ๐๐ ๐๐๐ ๐๐ ๐ = 10
๐๐จ๐ญ๐: โ๐
๐โ๐ ๐๐๐๐๐๐ ๐๐๐ ๐๐ ๐๐ข๐๐ ๐ ๐๐๐๐ = dim 0 = ๐ง๐๐๐
๐ผ๐ ๐ = 1,0,0 , 0,1,0 , (0,0,1) ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐๐ ๐3(๐น)
๐ท๐๐๐๐๐ ๐๐๐ ๐๐๐3(๐น) = dim ๐3(๐น) = ๐ ๐ = ๐๐. ๐๐ ๐๐๐๐๐๐๐ก๐ ๐๐ ๐๐๐ ๐๐ ๐ = 3
24. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
5. DIMENSION OF A SUBSPACE
Definition : DIMENSION OF A SUBSPACE
Let V F be the finite dimensional vector space and W F be the subspace of V F
The Number of elements in any basis of W F is called the dimension of W
and denoted by dim W
๐ฟ๐๐ก ๐ = ฮฑ1, ฮฑ2, ฮฑ3, โฆ , ฮฑ7 ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐๐ ๐(๐น)
For Example :-
๐ท๐๐๐๐๐ ๐๐๐ ๐๐ ๐ ๐น = dim ๐ = ๐ ๐ = ๐๐. ๐๐ ๐๐๐๐๐๐๐ก๐ ๐๐ ๐๐๐ ๐๐ ๐ = 7
๐ผ๐ ๐ = 1,0,0 , 0,1,0 , (0,0,1) ๐๐ ๐กโ๐ ๐๐๐ ๐๐ ๐๐ ๐3(๐น)
= dim ๐3(๐น)= ๐ ๐ = ๐๐. ๐๐ ๐๐๐๐๐๐๐ก๐ ๐๐ ๐๐๐ ๐๐ ๐ = 3
32. MANIKANTA SATYALA || LINEAR ALGBRA || BASIS AND DIMENSIONS
COSET PROPERTIES
1. for 0 โ V, 0 + W = W
โด W is itself a coset in V, generated by 0
2. for ๐ฅ โ W, ๐ฅ + W = W
โด Coset ๐ฅ + W = Coset W
4. if ฮฑ + W and ฮฒ + W are two cosets of W in V then
๐ผ + ๐ = ๐ฝ + ๐ โ ๐ผ โ ๐ฝ โ ๐
3. any two cosets of W in V are either identical or disjoint
๐. ๐. , ๐ธ๐๐กโ๐๐ ๐ผ + ๐ = ๐ฝ + ๐, ๐๐ ๐ผ + ๐ โฉ ๐ฝ + ๐ โ ฯ
6. QUOTIENT SPACE