Search results

1. Prove that T(G) is subgroup of G

Homework Statement Homework Equations subgroup axioms: 1. a, b in T(G), then ab in T(G) 2. existence of identity element. 3. a in T(G), then a^-1 in T(G) The Attempt at a Solution 1. let a be in T(G), then a^n = e. let b be in T(G), then b^n = e (ab)^n = (a^n)(b^n) = (e)(e) = e axiom 1...
2. Quickie: vector normal to surface

Homework Statement Find vector normal to z = x^2 + y^2 - 3 at point r = (2, -1, 2) Homework Equations The Attempt at a Solution here is the markscheme. I understand how to find the gradient, but i dont understand how they calculated the magnitude. thanks
3. Vector space, linear transformations & subspaces

Homework Statement Let V be a vector space over a field F and let L and M be two linear transformations from V to V. Show that the subset W := {x in V : L(x) = M(x)} is a subspace of V . The Attempt at a Solution I presume it's a simple question, but it's one of those where you just don't...
4. Simple matrix/linear algebra question, help

Homework Statement Here is the question, i know how to do part (i) but I do not understand part (ii): The Attempt at a Solution [/B] here's the solution from the marking scheme: i understand how they formed the matrix from their working out (i can se the pattern), but I do not...
5. Quick directional derivative question -- help please

Homework Statement [/B] find directional derivative at point (0,0) in direction u = (1, -1) for f(x,y) = x(1+y)^-1 The Attempt at a Solution grad f(x,y) = ( (1+y)^-1, -x(1+y)^-2 ) grad f(0,0) = (1, 0) grad f(x,y) . u = (1,0).(1,-1) = 1. seems easy but markscheme says im wromg. It says...
6. Lagrange multipliers, guidance needed

Homework Statement f(x,y) is function who's mixed 2nd order PDE's are equal. consider k_f: determine the points on the graph of the parabloid f(x,y) = x^2 + y^2 above the ellipse 3x^2 + 2y^2 = 1 at which k_f is maximised and minimised. The Attempt at a Solution is this the...
7. Partial derivatives

Homework Statement let w(u,v) = f(u) + g(v) u(x,t) = x - at v(x,t) = x + at show that: \frac{\partial ^{2}w}{\partial t^{2}} = a^{2}\frac{\partial ^{2}w}{\partial x^{2}} The Attempt at a Solution w(x-at, x+at) = f(x-at) + g(x+at) \frac{\partial }{\partial t}(\frac{\partial...
8. Field axioms, subspaces

Homework Statement Let F_{2} = {0, 1} denote a field with 2 elements. Let V be a vector space over F_{2}. Show that every non-empty set W of V which is closed under addition is a subspace of V. The Attempt at a Solution subspace axioms: 0 elements, closed under scalar multiplication...
9. Directional derivative Q, stuck.

Homework Statement In what directions at the point (2, 0) does the function f(x, y) = xy have rate of change -1? D_{u}(f)(a,b) = \bigtriangledown f(a,b)\cdot (u_{1}, u_{2}) f(x,y) = xy (a,b) = (2,0). The Attempt at a Solution \frac{\partial f}{\partial x} = y \frac{\partial...
10. Abelian groups, vector spaces

∴Homework Statement Let ℝ>0 together with multiplication denote the reals greater than zero, be an abelian group. let (R>0)^n denote the n-fold Cartesian product of R>0 with itself. furthermore, let a ∈ Q and b ∈ (ℝ>0)^n we put a⊗b = (b_1)^a + (b_2)^a + .... + (b_n)^a show that the...
11. Directional derivative, help

Homework Statement D_{u}(f)(a,b) = \triangledown f(a,b)\cdot u D_{(\frac{1}{\sqrt2}, \frac{1}{\sqrt2})}(f)(a,b) = 3 \sqrt{2} where u = (\frac{1}{\sqrt2}, \frac{1}{\sqrt2}) find \bigtriangledown f(a.b) Homework Equations The Attempt at a Solution first you change grad f into it's partial...
12. Vector space, abelian groups

Homework Statement The set ℝ^2 with vector addiction forms an abelian group. a ∈ ℝ, x = \binom{p}{q} we put: a ⊗ x = \binom{ap}{0} ∈ ℝ^2; this defines scalar multiplication ℝ × ℝ^2 → ℝ^2 (p, x) → (p ⊗ x) of the field ℝ on ℝ^2. Determine which of the axioms defining a...
13. Vector space and fields question

Homework Statement Let V be a vector space over the field F. The constant, a, is in F and vectors x, y in V. (a) Show that a(x - y) = ax - ay in V . (b) If ax = 0_V show that a = 0_F or x = 0_V . Homework Equations axiom 1: pv in V, if v in V and p in F. axiom 2: v + v' in V if v, v' in V...
14. Bayes theorem, answer way too small something wrong?

im sure i followed it correctly but my answer is unusually small... 1 in a thousand people have a disease. A company has discovered a new method for testing for the disease. If a person has the disease, the test will return a +ve result 99% of the time. If a person doesn't have the disease...
15. General probability question

Shortly after being put into service, some buses manufactured by a certain company have developed cracks on the underside of the main frame. Suppose a particular city has 25 of these buses, and cracks have actually appeared in 8 of them. (i) How many ways are there to select a sample of 5 buses...
16. No cycles in permutation N how to calculate sgn(N^2)?

N is a 2 x n matrix: N = 1 2 3 4 ... n-1 n n n-1 .... 4 3 2 1 then N^2 = 1 2 3 4 ... n-1 n 1 2 3 4 ... n-1 n You COULD use the theorem: sgn(N^2) = sgn(N)sgn(N) however, im asked to find sgn(N^2) by the traditional method: sgn(N^2) = (-1)^([L1 - 1] + [L2 - 1].....) where L represents the...
17. Square of a permutation matrix

say i have the matrix (4,2,5,6,3,1) and on top I have (1,2,3,4,5,6) i.e. a 2x6 permutation matrix. Let's call it sigma. how would I calculate (sigma)^2? I can break it down into cycles: sigma = <1,4,6>compose<3,5> thanks.