Give an example of a vector in R3 which does not belong to t

Give an example of a vector in R^3 which does not belong to the set W = { ( a, a+b, 2a - 3b ) } , a and b belong to R?

Solution

W is given as {(a, a+b, 2a-3b)} where a, b belong to real numbers.

We have to find an example of a vector in R^2 which does not belong to set W.

In other words, it is sufficient to select ordered triple (x,y,z) which do not satisfy the conditions of W

So let a = 1, b =1

Then select a vector as {(a,a-b, a+b)} = (1,0,2)

This obviously does not satisfy a =1, a+b =0 2a-3b = 2

For if a =1, then a+b=0 give b =-1

2a-3b = 2+3 =5

Hence this is one example (1,0,2) which is not in W.