Question

Check whether the set of vectors <a1, a2, 0> is a vector space. a1, a2 belongs to R​

Answer 1

Given:

The set of vectors <a1, a2, 0>  and

a1 , a2 belongs to R

To Find:

Check whether the set of vectors <a1, a2, 0> is a vector space.

Solution:

For any set of vector to form a vector space, the vectors should be linearly independent.

Consider 3 vectors V1, V2 and V3.

Then V1, V2 and V3 are linearly independent if ,

• mV1 + nV2 + pV3 = 0 ,
• only when a =0, b= 0 and c = 0.

In the question, the vectors are a1, a2 and 0.

• a1 and a2 are real numbers.

For <a1,a2,0> to be linearly independent,

• m(a1) + n(a2) + p(0) = 0 , for m=0 , n=0 and p=0

But its clear that p can be any number.

Also a1, a2 are real numbers,

• m a1 = -n a2
• a1 = -a2 x n/m

This is true for m ≠ 0

Therefore the given set of vectors <a1,a2,0> is not a vector space.