Identify if the following statements are True or False Be su

Identify if the following statements are True or False. Be sure to justify each answer. If {u, v} is linearly independent then so is {u, u + v}. If {u, v} is linearly independent then so is {u, v, u + v}.

Solution

(a) Let u and v be linearly independent. Also,let u and u +v be linearly dependent Then there exist non-zero scalars p and q such that pu + q(u+v) = 0 or, (p+q)u + qv = 0. Since u and v are linearly independent, this implies that p+q = 0 and q = 0 so that p = -q is also equal to 0 which is a contradiction. Hence u and u+v are linearly independent. The statement is TRUE.

(b) Since u + v -(u+v) = 0 , it implies that a linear combination of u, v and u+v is 0 without the coefficients being all 0. This implies that the vectors u, v and u+v are linearly dependent regardless of u, v being linearly independent. The statement is FALSE.