Page 80 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes 3
e
5. If e , ,e is a basis for R , then show that
1 2 3
e ,e e ,e e is also a basis.
2 3 1 1 2
6. Show that the set S 1,0,0 1,1,0 0,1,1 , 0,1,0 spans V R , but does not form a basis.
3
7. Show that the set 2, 1,0 3,5,1 1,1,2 forms a basis of V R .
3
3.3 Summary

 Let V F be a vector space and let S u 1 ,u 2 ,...u n be a finite subset of V . Then S is said to

be linearly dependent if there exists scalars , ... , F not all zero, such that
1 2 n
u u ... u 0.
1 1 2 2 n n
 Let V F be a vector space and let S u ,u ,...u be finite subset of V. Then S is said to be
1 2 n
linearly independent if

n
a u i 0, 1 . F
u
i 1
n
holds only when 0, i 1,2,... .
i
3.4 Keywords

Dimension: The Dimension of a finite space V over F is thus the number of elements in any basis
of V over F.

Linear Combination: V (F) is generated by the unit vectors (1,0,0), (0,1,0), (0,0,1) therefore that
3
elements of V (F) is a linear combination of the given set S.
3
3.5 Review Questions

1. Prove that a set of vectors containing null vector is a linearly dependent set.
x
2. Prove that the three functions ,cos and t 2 x e are linearly independent.
3. Prove that the set (1,2,0)(2,1,2)(3,1,1) is a basis for R .
3
4. Prove that if two vectors are linearly dependent, one of them is a scalar multiple of the
other.

3.6 Further Readings

Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I N Herstein, Topics in Algebra

Michael Artin, Algebra

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