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...
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
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...
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...
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...
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...
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...
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...
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...
∴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...
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...
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...
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...
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...
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...
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...
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.