Reference no: EM133354173
Question 1: Consider the linear system of equations:
x1 + 2x2 + x3 = -α
x1 + 3x2 + 2x3 = 1
3x1 + 7x2 + α2x3 = 3 - α
Find conditions on α ∈ R such that the system has no solutions, a unique solution, and infinitely many solutions. In the latter two cases, find the solution set in R3×1. (Note. Your solutions may be in terms of α.)
Question 2. Define the linear transformation T : P2(R) → R3×1 as
![377_linear transformation.jpg](https://secure.expertsmind.com/CMSImages/377_linear transformation.jpg)
where p'(t) is the derivative of p(t) and p"(t) is the second derivative of p(t). As an example, T (3 + 2t + t2) = (11, 5, -2)T. Additionally, define
![1571_linear transformation1.jpg](https://secure.expertsmind.com/CMSImages/1571_linear transformation1.jpg)
as well as the standard bases
![244_linear transformation2.jpg](https://secure.expertsmind.com/CMSImages/244_linear transformation2.jpg)
(Note. Throughout your solutions, you may use without proof the fact that EP2(R) is a basis for P2(R) and ER3×1 and C are bases for R3×1.)
(a) Prove that B is a basis for P2(R).
(b) Find R(T ) and rank(T ).
(c) Find N(T ) and null(T ).
(d) Is T injective, surjective, both, or neither? Explain briefly.
(e) Find [T ]ER3×1 ,EP2(R). (f) Find [T]C,B.
Question 3. Let A, B ∈ Cn×n such that AB = 0. Prove or disprove each of the following statements:
(a) Either A = 0 or B = 0 (or both).
(b) BA = 0. (c) (BA)2 = 0.
(d) If tr(A) = -4, then B = 0.
(e) If det(A) = -4, then B = 0.
To disprove a statement, provide a counterexample for n = 2. Include a brief explanation confirming that your chosen matrices form a counterexample.
Question 4. Let B ∈ Cn×n be a specific fixed matrix. Define VB to be the set of matrices in Cn×n that commute with B, that is
VB = {A ∈ Cn×n : AB = BA}.
Prove that VB is a subspace of the complex vector space Cn×n (with the usual matrix addition and scalar multiplication).