Find an orthogonal basis for t that contains the vector

Question

Problems:

B1) This question concerns the following two subsets of i³:

                                  S = {(1,1,-1),(4,2,0)}
                                  T = {(2a+b,a,b):a,b∈i}

(a) Show that S⊆T, and find a vector in i³  that does not belong to T.

(b) Show that T is a subspace of i³

(c) Show that S is a basis for T, and write down the dimension of T.

(d) Find an orthogonal basis for T that contains the vector (1,1,-1)

(e) Express the vector (2a+b,a,b)of T as a linear combination of the vectors in your orthogonal basis for T.

B2 This question concerns the function t given by the rule

                t:i³→i³

                (a,b,.c)→(a+2b+2c,-a-2b,2a+3b-2c)

                and S  = {(1,0,0),(1,1,0),(1,1,1)} and U = {(1,0,0),(1,1,0),(1,0,1)}

(a) Use the strategy in Unit 4, section 1, to show that t is a linear transformation.

(b) Write down the matrix for t with respect to the standard basis in both the domain and codomain.

(c) Determine the matrix of t with respect to the domain basis S of i³ and the standard basis in the codomain.

(d) Determine the matrix of t with respect to the domain basis S of i³ basis and codomain basis of U.

(e) Find the kernel of t, and states its dimension.

(f) Let  m:i³→ i³be a linear transformation given by m(a,b,c)=(b,a,a-b). Determine the matrix of mot and with respect to the standard basis in both the domain and the codomain.