Page 88 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Self Assessment

1. Find the co-ordinate matrix for the vector 1,0,1 in the basis of C consisting of the
3
vectors 2 ,1,0 , 2, 1,1 , 0,1i i ,1 i in that order.

3
2. Let , , 3 be the ordered basis for R . Consisting of 1,0, 1 ,
1 2 1
1,1,1 , 1,0,0
2 3
What are the co-ordinates of the vector a , ,c in the above ordered basis .
b
3. Let R be the field of the real numbers and let be a fixed real number. Let the new basis
, be given in terms of the matrix P by the relation
1 2
cos sin
1 1
sin cos
2 2
Here 1 1,0 and 2 0,1
cos sin
P
sin cos

Find the co-ordinates of the vector x ,x in terms of the new basis .
1 2
4. Show that the set of vectors 1 , 2 , 3 given by

1,0,0
1
4,2,0
2
5, 3,8
3
3
,
form a basis of F . Find the co-ordinates of the vector x ,x x in the basis '.
1 2 3
4.3 Summary

 The dimension of a vector space is the number of basis vectors of the vector space V over
the field. The standard basis for a three dimensional vector space is taken as

l 1,0,0
1
l 2 0,1,0

l 3 0,0,1
and they form an independent set of vectors and span the whole V over the field R.
3
3
 The co-ordinates in the three dimensional space F are x, y, z co-ordinates. So the co-
ordinates of a vector in V relative to the basis will be the scalars which serve to express
as a linear combination of the vectors in the basis.

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