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Question 2. Give an example of the following, or if no example exists state that. As...
Question 2. Give an example of the following, or if no example exists state that. As always, prove your answer in e...
Question 2. Give an example of the following, or if no example exists state that. As always, prove your answer in either case. (a) A finite non-normal extension of Q (b) A finite non-normal extension of R (c) A finite non-normal extension of F7
Question 2. Give an example of the following, or if no example exists state that. As always, prove your answer in either case. (a) A finite non-normal extension of Q (b) A finite non-normal extension of...
Question 1 (15 points) (a) Determine whether the following extensions are normal. Justify your answer. (1)...
Question 1 (15 points) (a) Determine whether the following extensions are normal. Justify your answer. (1) Q CQ(V2, 92). (2) F = F3(t) C F(a), where t is a variable and a is a root of x3 – t in the splitting field. (b) Give an example of a normal extension of Q that is not finite. Justify your answer.
Give an example for each of the following, or explain why no example exists. (a) A...
Give an example for each of the following, or explain why no example exists. (a) A non-diagonalisable (square) matrix. (b) A square matrix (having real entries) with no real eigenvalues. (c) A 2 x 2 matrix B such that B3 = A where A = (d) A diagonalisable matrix A such that A2 is not diagonalisable.
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2....
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Abstract Algebra Provide and explain an example of the following, and state explicitly if no such...
Abstract Algebra Provide and explain an example of the following, and state explicitly if no such example exists A unit, except t1 in an Integral domain R of your choice, which is not a field.
Abstract Algebra Provide and explain an example of the following, and state explicitly if no such example exists A unit, except t1 in an Integral domain R of your choice, which is not a field.
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
Example Question Suppose a molecule exists as a one dimensional harmonic oscillator in a superposition state...
Example Question Suppose a molecule exists as a one dimensional harmonic oscillator in a superposition state that is given by the following wavefunction: 1 15 Y = -4 +291 Where Y. and Y, are the ground state and the first excited state wavefunctions of the harmonic oscillator. Evaluate the expectation value of the vibrational energy this molecule in such a superposition state (in cm ) given that the vibration constant for the molecule is about 1800 cm
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be se...
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable? (b) Give an example of a finite extension of fields which is inseparable. (c) Discuss the extent to which the concept of separability can be removed from a class in Galois theory.
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable?...
Give an example of each of the following, or state that such a request is impossible....
Give an example of each of the following, or state that such a request is impossible. (No proof is required.) d) A continuous function f : R R that maps a closed interval1, onto an open interval (-π, π), i.e., f( [-1, 1] ) = (-π, π).
Question 2. Give an example of the following, or if no example exists state that. As always, prove your answer in either case. (a) A finite non-normal extension of Q (b) A finite non-normal extension of R (c) A finite non-normal extension of F7
Question 2. Give an example of the following, or if no example exists state that. As always, prove your answer in either case. (a) A finite non-normal extension of Q (b) A finite non-normal extension of...
Question 1 (15 points) (a) Determine whether the following extensions are normal. Justify your answer. (1) Q CQ(V2, 92). (2) F = F3(t) C F(a), where t is a variable and a is a root of x3 – t in the splitting field. (b) Give an example of a normal extension of Q that is not finite. Justify your answer.
Give an example for each of the following, or explain why no example exists. (a) A non-diagonalisable (square) matrix. (b) A square matrix (having real entries) with no real eigenvalues. (c) A 2 x 2 matrix B such that B3 = A where A = (d) A diagonalisable matrix A such that A2 is not diagonalisable.
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
Abstract Algebra Provide and explain an example of the following, and state explicitly if no such example exists A unit, except t1 in an Integral domain R of your choice, which is not a field.
Abstract Algebra Provide and explain an example of the following, and state explicitly if no such example exists A unit, except t1 in an Integral domain R of your choice, which is not a field.
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
Example Question Suppose a molecule exists as a one dimensional harmonic oscillator in a superposition state that is given by the following wavefunction: 1 15 Y = -4 +291 Where Y. and Y, are the ground state and the first excited state wavefunctions of the harmonic oscillator. Evaluate the expectation value of the vibrational energy this molecule in such a superposition state (in cm ) given that the vibration constant for the molecule is about 1800 cm
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable? (b) Give an example of a finite extension of fields which is inseparable. (c) Discuss the extent to which the concept of separability can be removed from a class in Galois theory.
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable?...
Give an example of each of the following, or state that such a request is impossible. (No proof is required.) d) A continuous function f : R R that maps a closed interval1, onto an open interval (-π, π), i.e., f( [-1, 1] ) = (-π, π).
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