Reference no: EM133335328

Question 1. Let T : P³(R) → R be a linear transformation for which

T (1) = 1, T (1 + t) = 2, T (1 + t + t²) = 3, T (1 + t + t² + t³) = 4.
(a) Find T (1 + 3t + 4t² + 2t³).
(b) Find T (a + bt + ct² + dt³) for any a, b, c, d ∈ R.

Question 2. Let T : R⁴ → R⁴ be a linear transformation for which
T (a, b, c, d ) = (a + b, b + c, c + d , d + a)
(a) Find a basis for N(T ).
(b) Find a basis for R(T ).

Question 3. Prove the following parts of Theorem 2.26:
(a) (1) T is injective if and only if rank(T ) = dim(V ).
(b) (5) If dim(V ) < dim(W ), then T is not surjective.

Question 4. Define the following complex vector space

 

with the usual addition and scalar multiplication.

(a) Find an appropriate value of r such that V ≈ C^r.
(b) Give an explicit isomorphism T : V → C^r. Make sure to prove that T is an isomorphism!

Question 5. Let A, B ∈ F^n×n be invertible matrices.
(a) Prove that AB is invertible with (AB)^-1 = B^-1A^-1.
(b) Prove that (A^T)^-1 = (A^-1)^T.