Page 86 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes or c c 0
1 3
c 0
1
c 2 0
c 4c 2c 0
2 3 4
So we get c 0
1
c 2 0

c 0
3
c 0
4
'
'
'
Thus the four set of vectors ' , , , are independent. Let P be a matrix such that
1 2 3 4
'
1 1
'
2 P 2
'
3 3
' 4
4
1,0,0,0
1
0,1,0,0
where 2
3 0,0,1,0
0,0,0,1
4

1 1 0 0
0 0 1 1
So P 1 0 0 4
0 0 0 2

Let P = –2, so, P is non-singular and invertible.

0 0 1 2
1 0 1 2
P 1
0 1 0 1/2
0 0 0 1/2

'
Thus ' – 2 , ' – ' 2 '
1 3 4 2 1 3 4
' – ' , '
3 2 4/2 4 4/2 is the answer.

Example 3: Let V be the vector space over the complex numbers of all functions from R
into C i.e. the space of all complex-valued functions on the real line. Let
f x 1, f x e ix , f 3 x e 1x .
2
1
f
(a) Prove that f , , f 3 are linearly independent.
1 2

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