4. Assume that A, B E Mnxn(R). Prove or disprove each of the following statements. (a)...
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix, 5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
Prove or disprove the following. (a) R is a field. (b) There is an additive identity for vectors in R^n. (If true, what is it?)........ 1. Prove or disprove the following. (a) R is a field (b) There is an it?) additive identity for vectors in R". (If true, what is (c) There is a is it? multiplicative identity for vectors in R". (If true, what (d) For , , (e) For a, bE R and E R", a(b) =...
Prove or disprove the following statements. In each case, A and B are both nx n matrices. (a) If C is a 3 x 2 matrix, then C has a left inverse. (b) If Null(AT) = {Õ}, then A is invertible. (c) If A and B are invertible matrices, then A + B is invertible.
2) Prove or disprove the following statements: (a) "If A E M5(R) has a non-real eigenvalue, then A is diagonalizable." (b) "If 2 E C" is an eigenvector of A € Mn(C) then Z is also an eigenvector of A."
For any matrix A E Mnxn(R), define a function TA: Mnxn(R) → Mnxn(R) by the formula TA(B) = AB + tr(A)] where Be Mnxn(R), tr(A) is the trace of A, and I e Mnxn(R) is the identity matrix. Under what conditions on A is TA a linear transformation that is also an isomorphism?
2) Prove or disprove the following statements: (a) “If A € M5(R) has a non-real eigenvalue, then A is diagonalizable." (b) “If z EC” is an eigenvector of A E Mn(C) then 2 is also an eigenvector of A.”
Let A. B, C, D є Mnxn(F), and det(A) 0, AC-CA. Prove that A B det ( )) -det(AD CB)
4. (15 pts) Suppose that R and S are reflexive relations on a set A. Prove or disprove each of these statements. (Note that RI R2 consists of all ordered pairs (a, b), where student a has taken course b but does not need it to graduate or needs course b to graduate but has not taken it.) a) R U S is reflexive. b) R S is reflexive. c) R田s is irreflexive. d) R- S is irreflexive. e) S。R...
Topology 3. Either prove or disprove each of the following statements: (a) If d and p map (X, d) X, then the identity topologically equivalent metrics (X, p) and its inverse are both continuous are two on (b) Any totally bounded metric space is compact. (c) The open interval (-r/2, n/2) is homeomorphic to R (d) If X and Y are homeomorphic metric spaces, then X is complete if and only if Y is complete (e) Let X and Y...
Prove or disprove the following expression. (Prove: using Boolean algebra. Disprove: using truth table.) (NOT is presented by -.) 1. a + b (c^- + d)^- = a^-b^- + a^-cd^- 2. ab^- + bc^- + ac^- = (a + b + c) (a^- + b^-+ c^-) 3. a^- + bd^-^- (c + d) + ab^-d = ac^-d + ab^-cd + abd