Math 233
Complete the following exercises.
1. Find ker(T) of T:M2×2−→ M2×2is the linear transformation defined by
T a b
c d != c−2a2b
0a−d+c!.
2. Consider the set of 2 ×2 matrices
S=( 1 1
0 0 !, 0 1
0 1 !, 1 1
0 1 !, 3 2
0 0 !).
(a) Is Sa linearly independent set?
(b) Find a basis Bfor span(S).
(c) What is the dimension of span(S)?
(d) Write the matrix A= 6 2
0 1 !as a linear combination of the vectors in the
basis B.
(e) What is the coordinate vector [A]B?
(f) For what value of kis [A]B∈Rk?
(g) Let Cbe the standard basis of M2×2. For what value of kis [A]C∈Rk? What is
[A]C?
3. Let Vbe a subspace of Pof polynomials of the form p(x) = ax4+bx3+cx2+dx +e
such that a−c= 0 and e−3b= 0.
(a) Find a basis for V.
(b) What is the dimension of V?
1