Question

Let G be the subgroup of R^3 consisting of all vectors of the form (x, y,...

Let G be the subgroup of R^3 consisting of all vectors of the form (x, y, 0). Let G act on R^3 by left multiplication. Describe the orbits of this G-action geometrically. Show that the set of orbits are in one to one correspondence with R

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Answer: It is given that

The action of on is defined as

Now, for a fixed , the orbit of , by definition, is

which is same as the set

and geometrically it is the plane parallel to the plane and intersecting the axis at

Now, let be the set of all orbits of the action on and define as map as for any It follows from the computation of the last paragraph that is a well defined bijection and hence the set of orbits is in one to one correspondence wtih

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let W be the subset of R^R consisting of all functions of the form x ?→a...
Let W be the subset of R^R consisting of all functions of the form x ?→a · cos(x − b), for real numbers a and b. Show that W is a subspace of R^R and find its dimension.
Exercise 9.1.11 Consider the set of all vectors in R2,(x, y) such that x + y...
Exercise 9.1.11 Consider the set of all vectors in R2,(x, y) such that x + y ≥ 0. Let the vector space operations be the usual ones. Is this a vector space? Is it a subspace of R2? Exercise 9.1.12 Consider the vectors in R2,(x, y) such that xy = 0. Is this a subspace of R2? Is it a vector space? The addition and scalar multiplication are the usual operations.
Show the vectors [x y z] where xyz=0 is a subspace V of R^3. is it...
Show the vectors [x y z] where xyz=0 is a subspace V of R^3. is it closed under additon? is it closed under scalar multiplication?
Let G be a non-trivial finite group, and let H < G be a proper subgroup....
Let G be a non-trivial finite group, and let H < G be a proper subgroup. Let X be the set of conjugates of H, that is, X = {aHa^(−1) : a ∈ G}. Let G act on X by conjugation, i.e., g · (aHa^(−1) ) = (ga)H(ga)^(−1) . Prove that this action of G on X is transitive. Use the previous result to prove that G is not covered by the conjugates of H, i.e., G does not equal...
1. Let W be the set of all [x y z}^t in R^3 such that xyz...
1. Let W be the set of all [x y z}^t in R^3 such that xyz = 0. Is W a subspace of R^3? 2. Let C^0 (R) denote the space of all continuous real-valued functions f(x) of x in R. Let W be the set of all continuous functions f(x) such that f(1) = 0. Is W a subspace of C^0(R)?
(a) Consider x^2 + 7x + 15 = f(x) and e^x = g(x) which are vectors...
(a) Consider x^2 + 7x + 15 = f(x) and e^x = g(x) which are vectors of F(R, R) with the usual addition and scalar multiplication. Are these functions linearly independent? (b) Let S be a finite set of linearly independent vectors {u1, u2, · · · , un} over the field Z2. How many vectors are in Span(S)? (c) Is it possible to find three linearly dependent vectors in R^3 such that any two of the three are not...
Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0...
Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show that there is a real number c such that f(x) = cg(x) for all x. (Hint: Look at f/g.) Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be the line tangent to the graph of g that passes through the point...
Let H be a subgroup of the group G. Define a set B by B =...
Let H be a subgroup of the group G. Define a set B by B = {x ∈ G | xax−1 ∈ H for all a ∈ H}. Show that H < B.
Let R[x] be the set of all polynomials (in the variable x) with real coefficients. Show...
Let R[x] be the set of all polynomials (in the variable x) with real coefficients. Show that this is a ring under ordinary addition and multiplication of polynomials. What are the units of R[x] ? I need a legible, detailed explaination
(a) Let the statement, ∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y). For each of the following...
(a) Let the statement, ∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y). For each of the following statements, decide if the statement is certainly true, certainly false,or possibly true, and justify your solution. 1 (i) G(3,4) (ii) ∀x∈RG(x,3) (iii) ∃y G(3,y) (iv) ∀y¬G(3,y)(v)∃x G(x,4)
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT