Page 89 - DMTH502_LINEAR_ALGEBRA

Unit 4: Co-ordinates

 If V is a finite-dimensional space, the ordered basis for V is a finite sequence of basis Notes
vectors , , ,... n which is a linearly independent set and spans V. So we just say that
1 2 3
, , ,...
1 2 3 n
is an ordered basis for V.

4.4 Keywords

,
n-tuple z z 2 ,...z : V is the n-tuple of co-ordinates of some vector z in V namely the vector
n
1
n
n
n
z z i i
i 1
Unique Scalars: P are such that
ij
n
' P j 1,2,...n
ij ij i
i 1

4.5 Review Questions

1. Show that the vectors
1,1,0,0 , 1,0,0,4 ,
1 2
0,0,1,1 , 0,0,0,2
3 4
4
form a basis for R . Find the co-ordinates of the standard basis vectors in the ordered basis
, , , .
1 2 3 4
2. Let W be the subspace of C spanned by 1,0, and 1 i ,1, 1
2
i
1 2
(a) Show that 1 and 2 form basis for W.
i
(b) Show that the vectors 1,1,0 and 1, ,1 i are in W and form an other basis
1 2
for W.
(c) What are the co-ordinates of 1 and in the ordered basis 1 , 2 for W?
2
Answers: Self Assessment

1 2 3 i
i
1. 1, ,
2 4
2. b c , ,a 2b c
b
x 1 ' cos x 1 sin x 2
3.
x ' –sin x cos x
2 1 2

11x x 3x x
4. x 1 2x 2 3 , 2 3 ' 3
8 2 8 8

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