1. For each of the following statements indicate if it is true or false. If the answer is
false, briefly explain why.

(a) (2 points) Let V be a vector space and consider the subspace W = Span{v1, v2, v3, v4}.
If v1 = 2v2 + v3, then {v2, v3, v4} is a basis for W.

(b) (2 points) If A and B are invertible n × n matrices, then A is row equivalent to B.

(c) (2 points) Let P3 be the vector space of all polynomials of degree less than or equal
to 3. If {p1, p2, p3} are linearly independent in P3, then they are a basis of P3.

(d) (2 points) If A is an invertible n × n matrix, then det(A3) > 0.

Find the number of r-permutations of the multiset {∞?1, ∞?2, … , ∞??} such that in
every such permutation each type of an element of the multiset appears at least
once. (You do not need to provide a short answer. Assume r ≥ n.)

(8) Suppose T : R 4 → R 4 with T(x) = Ax is a linear transformation such that • (0, 0, 1, 0) and (0, 0, 0, 1) lie in the kernel of T, and • all vectors of the form (x1, x2, 0, 0) are reflected about the line 2x1 − x2 = 0.

(a) Compute all the eigenvalues of A and a basis of each eigenspace.

(b) Is A invertible? Explain.

(c) Is A diagonalizable? If yes, write down its diagonalization (you can leave it as a product of matrices). If no, why not?

Linear Algebra: Explain what a vector space is and offer an example that contains at least five (5) of the ten (10) axioms for vector spaces.

a. Consider a non-equilateral triangle. Try to create a tessellation around a point as you did before. Be sure you have no gaps or overlaps. Do you think any triangle with tessellate? Why or why not? Defend your reasoning.

Decide, with justification, on the truth of the following propositions, both when the Universe of discourse is the set of all positive integers, and when the Universe of discourse is the set of all real numbers.

1.18. ∃x∀y,x≤y.
1.19. ∀y∃x,x≤y.
1.20. ∃x∀y,x<y.
1.21. ∀y∃x,x<y.
1.22. ∃x ∀y, y ≤ x.
1.23. ∀y ∃x, y ≤ x.
1.24. ∃x ∀y, y < x.
1.25. ∀y ∃x, y < x.
1.26. ∃x∀y,(x < y ⇒ x2 < y2). 1.27. ∀y∃x,(x<y⇒x2 <y2).

If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

user A sends two messages to user B using ElGamal. user C listens on the line and intercepts both messages, and got the decryption of the first message. explain well if user C can decrypt the second message as well:

1)user A sends two messages E(m) and E(m'). to encrypt the first message user A used the variable k and to encrypt the second k+5. User C knows that there's a difference of 5 between the k values chosen. can he recover the second message?

2)user A sends two messages E(m) and E(m'). to encrypt the first message user A used the variable k and to encrypt the second 2k. User C knows that the second k used is two times the first k used. can he recover the second message?

please explain mathematically and elaborate so i can understand and learn from your answer. thank you very much.

Two sets are separated if the intersection of the closure of one of the sets with the other is empty.

a) Prove that two closed and disjoint sets of some metric space are separate.

b) Prove that two open and disjoint sets of some metric space are separate.

Prove that there are real non-algebraic numbers. Give two examples, the two most famous examples.

2. Some fourth-grade students are practicing reading and comparing decimal numbers. They created two different numbers using decimal squares. Stephanie reads the decimal squares below as “zero and forty-five hundredths” and “zero and one tenth.” She says: “Zero and forty-five hundredths is thirty five hundredths greater than one tenth.” Her partner, Ingrid says: “That’s not right. The square on the left is thirty-five squares bigger than the one on the right.”

a) What does Stephanie seem to understand? What does Ingrid seem to understand?

b) What could you talk about with these students to help improve their understanding?

solve for x:

[x * sqrt(1+x²)] + ln[x + sqrt(1+x²)] = 25

Determine if the statement below is True or False. Justify your answer by giving a proof or counterexample. Let A,B,C∈Mn×n(R) . Suppose C is invertible and C=AB. Then the columns of A, B and C are each bases for Rn and B is the change of basis matrix from the columns of C to the columns of A.

Using Kurosch's subgroup theorem for free proucts,prove that every finite subgroup of the free product of finite groups is isomorphic to a subgroup of some free factor.