For matrices A ∈ Rn×n and B ∈ Rn×p, prove each of the following statements: (a)...

For matrices A ∈ Rn×n and B ∈ Rn×p, prove each of the following statements:
(a) rank(AB) = rank(A) and R(AB) = R(A) if rank(B) = n.
(b) rank(AB) = rank(B) and N (AB) = N (B) if rank(A) = n.

Expert Solution

Colby Messinger answered 2 years ago

Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2) +...

Prove that if A and B are 2x2 matrices, then (A + B)^(2) = A^(2) + AB + BA + B^(2). Hint: Write out 2x2 matrices for A and B using variable entries in each location and perform the operations on either side of the equation to determine whether the two sides are equivalent.

Prove the following statements! 1. If A and B are sets then (a) |A ∪ B|...

Prove the following statements! 1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|. 2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|. 3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b) injective but not surjective. (c) surjective but not injective. (d)...

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B +...

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B + CX) ?1 = C Write an expression for the matrix X in terms of A, B, and C. You may assume invertibility of any matrix when necessary. (b) Suppose D is a 3 × 5 matrix, E is a 5 × c matrix, and F is a 4 × d matrix. Find the values of c and d for which the statement “det(DEF) =...

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is...

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is equal to the number of partitions of n in which the two largest parts are equal.

All vectors are in R^ n. Prove the following statements. a) v·v=||v||2 b) If ||u||2 +...

All vectors are in R^ n. Prove the following statements. a) v·v=||v||2 b) If ||u||2 + ||v||2 = ||u + v||2, then u and v are orthogonal. c) (Schwarz inequality) |v · w| ≤ ||v||||w||.

Using the axioms of probability, prove: a. P(A U B) = P(A) + P(B) − P(A...

Using the axioms of probability, prove: a. P(A U B) = P(A) + P(B) − P(A ∩ B). b. P(A) = ∑ P(A | Bi) P(Bi) for any partition B1, B2, …, Bn.

Combinatorics Prove bell number B(n)<n!

*Combinatorics* Prove bell number B(n)<n!

Prove via induction the following properties of Pascal’s Triangle: •P(n,2)=(n(n-1))/2 • P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for all...

Prove via induction the following properties of Pascal’s Triangle: •P(n,2)=(n(n-1))/2 • P(n+m+1,n) = P(n+m,n)+P(n+m−1,n−1)+P(n+m−2,n−2)+···+P(m,0) for all m ≥ 0

Let W be a subspace of Rn. Prove that W⊥ is also a subspace of Rn.

Determine whether the following statements describe Cell Respiration (C), Photosynthesis (P), Both (B), or None (N)....

Determine whether the following statements describe Cell Respiration (C), Photosynthesis (P), Both (B), or None (N). a. Uses a proton gradient to create ATP. b. Uses byproducts to oxidize NADH to NAD+ for reuse. 2. The thylakoid space of the choloroplast is: a. More acidic than the stroma b. More basic than the stroma c. Hypotonic to the stroma d. At equilibrium with the stroma e. Full of electrons from the stroma

Question