Let Vi 105 V3 175 Determine whether the vectors V V2 V3 f

Let Vi = (1,0,5), V3 = ( 1.75) Determine whether the vectors V, V2, V3 form a basis for R Determine whether the vectors vi, v2, V3 form a basis for R3

Solution

A set of vectors are said to form the basis  if the following conditions are met.

1. The set spans all of their vector space.

2. The set is linearly independent.

So lets see if these vectors are linearly independent, so we have to see if

a(1,0,5) + b(2,1,4) + c(1,7,5) = 0

=> a+2b+c=0 (1)

b+7c=0 (2)

5a+4b+5c=0 (3)

Putting b=-7c from (2) in (1) and (3), we get

a+2(-7c)+c=0 and 5a+4(-7c)+5c=0 => a-14c+c=0 and 5a-28c+5c=0 => a-13c=0 and 5a-23c=0

Solving these two,putting a=13c in 5a-23c=0

5(13c)-23c=0=>65c-23c=0 =>42c=0 => c=0

Putting c=0 in (2)

b+7(0)=0 => b=0

Putting b=0 and c=0 in (1)

a+2(0)+0=0 => a=0

Since we have a=b=c=0, any set of three linearly independent vectors in R³ spans R³. Hence given set of vectors form basis for R³.