Page 111 - DMTH502_LINEAR_ALGEBRA

Unit 6: Computation Concerning Subspaces

 Given the vectors , row-reduction is a straightforward method of determining the integers Notes
i
r, k ,...,k and the scalars R which give the description of the subspace spanned by  ,..., .
1 r ij 1 m
 The question of whether  = (b ,...,b ) is a linear combination of the , and if so, what the
1 n i
scalars x are, can also be looked at by asking whether the system of equations
i
m
A x i b j , j 1,...,n
ij
i 1
has a solution and what the solutions are.

 The unit helps in finding an invertible matrix P such that the co-ordinates of a vector  in
the two system of basis  and ’ are related by the relation X = PX’ for every basis .

6.4 Keywords

Basis of the Subspace: The basis of the subspace W is found by the row vectors of R. So one can
test whether a vector  belongs to W or not.
Row Reduction of a Matrix: The row reduction of a matrix A helps whether a set of vectors  ,
1
 ,  form a basis by forming the matrix A with row vectors and finding its rank.
2 3
6.5 Review Questions

1. Let  = (u , u ,...,u ) and ’ = (v , v , v ,..,v ) be two bases of a vector space V. Show that the
1 2 n 1 2 3 n
base change matrix P is uniquely determined by the two bases  and ’ and is an invertible
matrix.
2. Solve completely the system of equations AX = 0 and AX = B, where

1  1 0   1 
   
A  1 0 1 and B   1
   
 1   1  1    1  
Answers: Self Assessment


 a b a b a b 





)
i

i

1.  (1 2 ), (1 2 )  (7i  1) ic , (3 i  (6 i ) c (1 i ) 
 5 5 5 5 5 
2. , 
6.6 Further Readings
Books Kenneth Hoffman and Ray Kunze, Linear Algebra
I.N. Herstein, Topics in Algebra
Michael Artin, Algebra

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