Question

(A) Prove that over the field C, that Q(i) and Q(2) are isomorphic as vector spaces?...

(A) Prove that over the field C, that Q(i) and Q(2) are isomorphic as vector spaces?

(B) Prove that over the field C, that Q(i) and Q(2) are not isomorphic as fields?

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
(A) Prove that over the field C, that Q(i) and Q(sqrt(2)) are isomorphic as vector spaces....
(A) Prove that over the field C, that Q(i) and Q(sqrt(2)) are isomorphic as vector spaces. (B) Prove that over the field C, that Q(i) and Q(sqrt(2)) are not isomorphic as fields
Consider vector spaces with scalars in the field F(could be R or C). Recall that L(V,...
Consider vector spaces with scalars in the field F(could be R or C). Recall that L(V, W) is the vectors space consisting of all linear transformations from V to W. a. Prove that L(F, W) is isomorphic to W. b. Assume that V is a finite dimensional vectors space. Prove that L(V, F) is isomorphic to V. c. If V is infinite dimensional, what happens to L(V, F)?
2. Determine whether the following pairs of vector spaces are isomorphic. If so, give an explicit...
2. Determine whether the following pairs of vector spaces are isomorphic. If so, give an explicit isomorphism (you must show that it is indeed an isomorphism); if not, state why not. a) R5 and P5 b) P8 and Mat33(R) c) {f ∈ P4 | f(2) = 0} and {(x1,...,x6) ∈ R6 | x1 = x4 = x5}
Let K be a field. Show that the category of vector spaces over K is an...
Let K be a field. Show that the category of vector spaces over K is an Abelian category.
Find a basis for each of the following vector spaces and find its dimension (justify): (a)...
Find a basis for each of the following vector spaces and find its dimension (justify): (a) Q[ √3 2] over Q (b) Q[i, √ 5] (that is, Q[i][√ 5]) over Q;
5.1.5. Suppose V1, V2, W are vector spaces over F. Prove that f : V1 ×...
5.1.5. Suppose V1, V2, W are vector spaces over F. Prove that f : V1 × V2 → W is the zero map if and only if f is both linear and bilinear.
For each of the following, determine whether it is a vector space over the given field....
For each of the following, determine whether it is a vector space over the given field. (i) The set of 2 × 2 matrices of real numbers, over R. (ii) The set of 2 × 2 matrices of real numbers, over C. (iii) The set of 2 × 2 matrices of real numbers, over Q.
Q 1 Determine whether the following are real vector spaces. a) The set C with the...
Q 1 Determine whether the following are real vector spaces. a) The set C with the usual addition of complex numbers and multiplication by R ⊂ C. b) The set R2 with the two operations + and · defined by (x1, y1) + (x2, y2) = (x1 + x2 + 1, y1 + y2 + 1), r · (x1, y1) = (rx1, ry1)
(a) Let a,b,c be elements of a field F. Prove that if a not= 0, then...
(a) Let a,b,c be elements of a field F. Prove that if a not= 0, then the equation ax+b=c has a unique solution. (b) If R is a commutative ring and x1,x2,...,xn are independent variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is isomorphic to R[x1,x2,...,xn] for any permutation σ of the set {1,2,...,n}
? is a vector field and ? is a scalar field 1. Prove in 3 dimensions:...
? is a vector field and ? is a scalar field 1. Prove in 3 dimensions: ∇ ∙ (∇ × ? ) = 0 2. Prove in 3 dimensions: ∇ × (∇φ) = 0 3. Prove in 3 dimensions: (?∙∇) ?= −? × (∇ × ?)+ ∇((?∙?)/ 2)