\f7) Given H = [(a,b,c):a - 3b + c = 0,b - 2c = 0,2b - c = 0] a) Find a basis for H. Show and explain why it is a basis. b) State the dimension of H. 6) Find the dimension of the subspace spanned by the given vectors. 8) Find the dimensions of Nul A and Col A for the matrices below: Explain using pivot columns of A. 1 -6 9 0 -2 a) A = 0 bj A = 1 2 -4 Lo 4 0 0 5 0 0 Dim Nul A = Dim Col A = Dim Nul A = Dim Col A = 9) Find the dimensions of Nul A and Col A for the matrices below: Explain using pivot columns of A. 10 0 1 -4 Dim Nul A = Dim Col A = 10) Determine True/False. Explain a) The number of variables in the equation Ax = 0 equals the dimension of Nul A. bj If dim V = n and if a set $ spans V, then 5 is a basis for V.1. Suppose you have vectors in R2 where the addition operation has been redened so that for any two vectors in R2, lo. . [:3 Jo. + 3c b of: i: 3d .1. ::u. Scalar multiplication is defined the usual way of c [:1 I J' i I J (5.} Proof that the above definition is not a vector space. 2. Let H be the set of polynomial functions divisible by i :1? EJ. Proof that H is a subspace of vector space ( is in H. closed under addition closed under scalar multiplication