ROB 501
Sample Midterm Questions
1. (Questions on logic and proof methods) Recall that ∧ is ‘and’, ∨ is ‘or’, and ¬ is ‘not’. Recall also that the symbol ⇔ and the written text, “if, and only if”, “logically equivalent to”, and “have the same truth table”, all mean the same thing. For example, in HW, you verified that ¬(p ∧ q) is “logically equivalent to” (¬p) ∨ (¬q) by proving “they have the same truth table”. Circle True or False as appropriate for the following state-ments:
T F (a) There are statements about the natural numbers that can be proved with Strong Induction but cannot be proved with Ordinary Induction.
T F (b) Let n be a natural number. If n 2 is odd then so is n.
T F (c) p =⇒ q is logically equivalent to (¬p) ∨ q
T F (d) The truth table given below is correct for ¬q =⇒ p
2. (Facts about matrices) For n, m ≥ 1, let A and B be an n × m real matrices. For any real matrix M, denote its i-th column by Mi and its ij-element by [M]ij . Circle True or False as appropriate for the following statements:
T F (a) trace(AAT) = ([A]ii)2
T F (b) [AT B]ij = (Ai) > Bj .
T F (c) span{A1, A2, · · · , Am} = {y ∈ R n | ∃x ∈ R m, such that y = Ax}.
T F (d) Suppose n = m so that A is a square real matrix and let x ∈ R n, x = [x1, x2, . . . , xn]T . Then
x1A1 + x2A2 + · · · + xnAn = xT A.
3. Let (X , R) be a finite-dimensional inner product space, let S1 ⊂ X and S2 ⊂ X be nonempty subsets (to be clear, they may or may not be subspaces). Circle True or False as appropriate for the following statements:
T F (a) If span{S1} ⊂ span{S2}, then S1 ⊂ S2.
T F (b) span{S1} ⊕ S⊥1 = X .
T F (c) span{S1 ∪ S2} = span{S1} + span{S2}.
T F (d) S⊥1 ∩ S⊥2 = [span{S1} ∩ span{S2}]⊥ .
4. (Normed spaces, matrices, and inner products) Circle True or False as appropriate for the following state-ments:
T F (c) The matrix M = is positive definite.
5. (Vector spaces, representations, and norms) Let (X , R, || · ||) be an n-dimensional normed space with n ≥ 4 and let L : X → X be a linear operator. Let A be the matrix representation of L : X → X when the basis {u} := {u1 , . . . , un} is used on both copies of X (i.e., on the domain of L and its range (also called co-domain)). We define a second basis on X by scaling the first basis:
Circle True or False as appropriate for the following statements, where u i and u¯ j always refer to elements of the given bases and the matrix A is as defined in the problem statement.
T F (a) The change of basis matrix from {u} to {u¯} (i.e. P ∈ R n×n s.t. [x]u¯ = P[x]u) is P = diag ([1, 2, . . . , n]).
T F (b) [L(u 3 )]{u¯} = 1 3A3, where A3 is the third column of A.
T F (c) [L(u¯ 3 + u¯ 4 )]{u} = 3A3 + 4A4 where A3 and A4 are the corresponding columns of A.
8. (15 points) (Proof Problem) Let (X , F) be a vector space and v 1 , v 2 , v 3 vectors in X . Define the following two statements:
• P: each of the sets {v 1 , v2}, {v 2 , v3}, and {v 3 , v1} is linearly independent.
• Q: the set {v 1 , v2 , v3} is linearly independent.
For each of the following statements, decide if it is T or F and then support your conclusion with a proof or a counterexample:
(a) P =⇒ Q
(b) Q =⇒ P
Show your work below. You can use as true anything we have established in ROB 501 lecture or HW. I cannot answer any question of the form. “do I have to prove this?” or “can I assume this?” or “have I shown enough?” .
Remark: If the problem seems completely trivial, that is OK; please write down the few lines it takes to do a (complete) proof or to establish a counterexample. If the problem seems challenging, then maybe you need more than a few lines to work the problem. Both are possible.
(a) P =⇒ Q is [T F (circle one)] and here is my supporting reasoning.
(b) Q =⇒ P is [T F (circle one)] and here is my supporting reasoning.
9. (5 points) A+ Problem: Points earned here will go toward deciding who goes from an A to an A+ at the end of the term. Recall that for your GPA at Michigan, an A+ counts the same as an A.
Def. Let (X , C) be a vector space over the complex numbers and L : X → X be a linear operator. λ ∈ C is an e-value of L if ∃ (v ∈ X , v = 0) such that L(v) = λv; v is called an e-vector of L.
Prove this: If λ1, λ2, and λ3 are distinct e-values of L, then a set of corresponding e-vectors {v 1 , v2 , v3} is linearly independent.
Note: If you need to use a result from lecture or HW, clearly state the result and note that it is from ROB 501; in that case, you do not need to prove it. Otherwise, any other statements used in your proof should be justified here. If your proof assumes that (X , C) is finite dimensional, you will earn at most three (3) points.