Page 87 - DMTH502_LINEAR_ALGEBRA

Unit 4: Co-ordinates

Notes
x
(b) Let g x 1,g x cos ,g x sin . Find an invertible 3×3 matrix P such that
x
1 2 3
3
g P f
j ij i for j 1,2,3
i 1
,
Solution: Let f x f x f x be a dependent set then we can find real , ,c c c not all of them
,
1 2 3 1 2 3
zero so that
c f x c f x c f x 0
1 1 2 2 3 3
or c 1 .1 c e ix c e ix 0 ...(1)
2
3
Taking real part we have
...(2)
c 1 c 2 cosx c 3 cosx 0
Taking imaginary part we have
...(3)
c c 0
2 3
From (2) we have c 1 0,c 2 c 3 0 for arbitrary x,

From (3) we have c 2 c 3
So we get c 1 c 2 c 3 .

,
,
which contradicts the statement that all c’s are not zero. So the set f f f is an independent set.
1
3
2
,
,
,
,
So find g g g in terms of f x f x f x we see that
3
1
2
3
1
2
g x f x 1
1 1
f x f x
g x cosx 2 3
2
2
f x f x
g x sinx 2 3
3
2i
g x 1 0 0 f x
1 1
Thus g x 0 1/2 1/2 f x ...(4)
2
2
g x 0 1/2i 1/2i f x
3 3
1 0 0
Thus P 0 1/2 1/2 ...(5)
0 i /2 /2
i
2i i
Also Let P 0 ...(6)
4 2
So P is invertible 3 × 3 matrix given by (5).

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