Determine the validity TrueFalse of the following statements

Determine the validity (True/False) of the following statements. Briefly justify each answer.. If {u, v} is linearly independent then so is {u + v, u - v}. Let A be a 8 times 7 matrix. If Null A consists of a plane then rank A = 6. Each line is a one-dimensional subspace of R^n. If the equation Ax = lambda x has a solution then lambda is an eigenvalue of A.

Solution

a) ( u, v ) are linearly independent then we have to show c1(u+v) + c2(u -v) =0

However u(c1+c2) +v(c1 - c2) =0

c1+c2=0 ;c1=c2

So, this is only possible when c1 = c2 =0

So, (u+v , u-v) are linerly indepdent . True

b) NulA is aplane which means NulA = 2 So, rank is not 6

So, False

c) False , line must go through (0,0)

d) True Ax = lamda*x