Page 78 - DMTH502_LINEAR_ALGEBRA

Linear Algebra

Notes Illustrative Examples

Example 5: Show that the set

S 1,2,1 2,1,0 , 1, 1,2 forms a basis for V (F).
3
a
Solution: Let a , ,a . F
1 2 3
then a 1,2,1 a 2,1,0 a 1, 1,2 0
1 2 3
a 2a a ,2a a a ,a 2a 0,0,0
1 2 3 1 2 3 1 3
a 1 2a 2 a 3 0,2a 1 a 2 a 3 0,a 1 2a 3 0
a 1 a 2 a 3 0.

Hence the given set is linearly independent.

Now let 1,0,0 x 1,2,1 y 2,1,0 z 1, 1,2

x 2y z ,2x y z ,x 2z

so that x 2y z 1,2x y z 0,x 2z 0

x 2/9,y 5/9,z 1/9
Thus, the unit vector (1,0,0) is a linear combination of the vectors of the given set, i.e.

(1, 0, 0) = –2/9 (1, 2, 1) + 5/9(2, 1, 0) + 1/9 (1, –1, 2)
Similarly,
(0, 1, 0) = 4/9 (1, 2, 1) – 1/9(2, 1, 0) – 2/9 (1, –1, 2) and

(0, 0, 1) = 1/3 (1, 2, 1) – 1/3(2, 1, 0) + 1/3 (1, –1, 2)
Since V (F) is generated by the unit vectors (1,0,0), (0,1,0),(0,0,1) we see therefore that ever
3
elements of V (F) is a linear combination of the given set S. Hence the vectors of this set form a
3
basis of V (F).
3
Example 6: Prove that system S consisting n vectors

e 1 1,0,...0 ,e 2 0,1,...,0 ...e n 0,0,...1 is a basis of V (F).
n
Solution: First we shall prove that the given system S is linearly independent.
Let a , a , ... a be any scalars, then
1 2 n
a e + a e + ... a e = 0
1 1 2 2 n n
a 1,0,...,0 a 0,1,...0 ... a 0,0,...,1 0
1 2 n
a 1 , ,...a n 0
a
2

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