Mapping Definition and 364 Threads

Homework Statement Show that the inversion mapping w = f(z) = 1/z maps the circle |z - 1| = 1 onto the vertical line x = 1/2. Homework Equations The Attempt at a Solution z = a + ib w = x + yi = a^2/(a^2 + b^2) + ib^2/(a^2 + b^2) |z - 1| = |(a -1) + ib | = 1 (a - 1)^2 + b^2 = 1...

How come the rank of a matrix is equal to the amount of pivot points in the reduced row echelon form? My book denotes this a trivial point, but unfortunately I don't see it :(

Homework Statement A- Find [L] for L : R3 → R3 where L first reflects throught the plane x−z = 0 and then rotates the zy-plane by pi/6 counterclockwise starting from the y-axis. B-Find [L] for L : R3 → R3 where L first rotates the xy-plane by pi/4 counterclockwise starting from the x-axis and...

Problem: Let L and M be finite dimensional linear spaces over the field K and let g: L\times M \rightarrow K be a bilinear mapping. Let L_0 be the left kernel of g and let M_0 be the right kernel of g. a) Prove that dim L/L_0 = dim M/M_0. b) Prove that g induces the bilinear mapping g': L/L_0...

Homework Statement Consider the mapping w = 1/(z-1) from the z-plane to the w plane. Show that in the z plane the circle (x-2)² + y² = 4 maps to a circle in the w-plane. What is the radius of this circle and where is it's centre. So in the z-plane this is a circle with radius 2 at the point...

Homework Statement here's the problem: Let A and B be n x n matrix with coefficient in K (any field), let Mn(K) be the set of all n x n matrix with coefficient in K . T is a linear map defined like this T : Mn(K)---> Mn(K) T(Y) = AYB what are the necessary conditions for T to be a...

[b]1. prove that,if T is acontraction mapping then T^n is a contraction Homework Equations The Attempt at a Solution

If a mapping between Lie algebras \varphi : \mathfrak{g} \to \mathfrak{h} takes a basis in \mathfrak{g} to a basis in \mathfrak{h} is it an isomorphism of vector spaces?

I want a sequence of n bits without all 1's in the output. What is the minimum number of bits k given n that can be used in any linear or non-linear invertible mapping that will produce such a sequence at output. For example, consider n=3. I want to create a mapping that has all 2^3=8 minus 111...

Homework Statement Show the map \varphi : \mathfrak{g} \to \mathfrak{h} defined by \varphi (aE + bF + cG) = \begin{bmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{bmatrix} is bijective. \mathfrak{g} is the Lie algebra with basis vectors E,F,G such that the following relations for...

so for a mapping f:X->Y where X,Y are Normed Vector Spaces if I have a function f(x) = y such that x in X and y in Y, how do I explicitly find f inverse? I sat down to do this and realize I've only been trained in the Reals where you switch the x,y and then solve for y. But this won't...

Homework Statement a) Let f: RN to RM. Define continuity for mapping f. How does this relate to the notion of metric (norm)? b) Define the Jacobian J of f. Write Taylor series expansion (for f) up to first degree at x = x0. Explain the terms. c) Let y = f(x) \in RM and yj = |f(x)|j = sum...

Hi, All: I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to SL(2,Z). The method: every homeomorphism h: <\tex> T^2 -->T^2<tex> gives rise to, aka, induces an isomorphism g: <\tex> \mathbb pi_1(T^2)-->\mathbb...

Hi . I am making a robot that is supposed to detect a red ball with a camera, and then know where the ball is . The robot has a camera on top of it, that is tilted at an angle ; At each frame the camera detects the red ball and returns 2 coordinates , xp and yp which are the pixel...

in some work I'm doing i have a 2D rectangle that can be rotated and/or translated in any direction in 2D space. for example it might look like this: (x=30,y=-10) +-----------+ (x=30,y=2) +-- +y | | |...

Hello, In class we're using the free course on complexe analysis by Ash & Novinger, legally downloadable online. I'm stuck on their proof of the open mapping theorem. More specifically: http://www.math.uiuc.edu/~r-ash/CV/CV4.pdf (page 15) proposition (d) is f(\Omega) \textrm{ is open} and its...

Dear Programmers I am having text file having huge data. From the text file i would like to map the two numbers that are present in the same brackets and the values should not considered if they are present in different numbers. say for example I want to map number "4758 & 3895" that are in...

I'm wondering why 1/k! is needed in Alt(T), which is defined as: \frac{1}{k!}\sum_{\sigma \in S_k} \mbox{sgn}\sigma T(v_{\sigma(1)},\cdots,v_{\sigma(k)}) After removing 1/k!, the new \mbox{Alt}, \overline{\mbox{Alt}}, still satisfies...

It is known that any linear mapping between two finite dimensional normed vector space is continuous (bounded). Can anyone give me an example of a linear mapping between two infinite dimensional normed vector space that is discontinuous? Thanks

Homework Statement Hello! I am stuck, having wondered about this question for quite some time now and I am not too sure how to solve it Denote the xy-plane by P. Let C be some general curve in P defined by the equation f(x,y) = 0, where f(x,y) is some algebraic expression involving x...

The function is one-one find the value of b^2+c^2[f(x)=x^3+3x^2+4x+bsinx+ccosxI tried the approach of monotonocity as a one one function will be strictly inc or decreasing in it's domain but I'm not being able to figure out b^2+c^2

Homework Statement Denote the xy plane by P. Let C be some general curve in P defined by the equation f ( x , y ) = 0 where f ( x , y ) is some algebraic expression involving x and y. Verify carefully that if B : P -> P is any bijection then B( C ) is defined by the equation f ( B^-1...

What I'm trying to do is to apply conformal mapping and map the area bounded by the x-axis and a line at 60 degrees to the x-axis to the region above the x-axis. I think the basic goal of what I'm trying to do is to map \pi/3 to \pi. My problem is I really have no idea where to go from there...

Hi all How to calculate the capacitance of a sphere-plane system by using conformal mapping? Thanks

Hello! I am trying to transform a circle into a "quarter moon shape". This is that every point in the circle is mapped into the "quarter moon shape" - therefore squeezed. In particular I am looking for a expression that relates x x' and y y' ... Can any bright mind help me :) ? Thank you

In a book I'm reading, it defines a bounded bilinear mapping \omega: X\times Y\rightarrow W, where X,Y and W are all normed linear spaces as \left\| \omega(\xi,\eta)\right\| \leq b \left\| \xi \right\| \left\| \eta \right\| So it uses \left\| \xi \right\| \left\| \eta \right\| as a norm on...

Hello! Please I need some help with this: Is it possible to transform a circle into a rectangle? If so what would be the expressions of x' and y' in terms of x and y. Thank you in advance!

In this context, the support mapping of any convex geometry is any point on the geometry which results in the largest dot product to some direction vector. I would appreciate some help in computationally finding the support mapping of an arbitrary ellipsoid (some arbitrary orthonormal basis...

Find any fixed points for following mapping f(z) = z2 - z + 1 Map onto the same point gives: z = z2 - z + 1 0 = z2 - 2z + 1 Therefore z = 1 and z = 1

Hello, I have an upcoming exam on human genetics and genetic techniques and I am trying to learn about this procedure. I have searched the internet but come up with more studies that have used this technique rather than a plain 'for dummies' explanation of what it is. If someone could offer...

Firstly, Hi everyone :) Secondly, please forgive my virtualy non existent knowledge of any theories that this relates to. I'm not a Dr,MD, or even a graduate. I'm not even studying anything anywhere. I have a few 'problems' (chronic anxiety, depression, yada yada yada) not looking for sympathy...

Hi, Everyone: I am reading a paper that refers to a "natural surjection" between M_g and the group of symplectic 2gx2g-matrices. All I know is this map is related to some action of M_g on H₁(S_g,Z). I think this action is...

Homework Statement We want to create a map from (x,y) to (u,v) such that the right side (positive x) of the hyperbola x^2 - y^2 = 1 is mapped onto the line v = 0 AND all the points to the left of that hyperbola are mapped to above the line. The mapping should be one-to-one and conformal...

Homework Statement Let f and g be bijective holomorphic maps from an open set A to the unit circle. Let a \in A and c=f(a) and d=g(a). Find a relation between f and g that involves a,c,d,f'(a),g'(a).Homework Equations The Attempt at a Solution If we also assumed that the open set is...

Solved, made a silly mistake. Thanks for reading.

1. Suppose that Ø:Z(50)→Z(50) is an automorphism with Ø(11)=13. Determine a formula for Ø(x). this is the problem I am getting, its chapter 6 problem 20 in Gallian's Abstract Algebra latest edition (you can find it on googlebooks) Am i wrong in thinking there's something wrong with the problem...

Reference: http://en.wikipedia.org/wiki/Bijection,_injection_and_surjection Consider the two sets X & Y connected by a the relation y^2=x^2. (For simplicity we can take X={-2,2} and Y={-2,2}).Then can we call the mapping from X to Y to be surjective? From the definition of WIKI, the answer...

Show that the mapping Phi(a+bi)=a-bi is an automorphism of the group of complex numbers under addition. I have this as of now: Let (c+di) and (a+bi) be elements in group G. 1) phi(a+bi) is function phi from G to G, by assumption. Therefore the function is a mapping. 2) 1-1: assume...

Homework Statement Let P2 be the vector space of real polynomials of degree less or equal than 2. Define the (nonlinear) function E : P2 to R as E(p)=integral from 0 to 1 of ((2/pi)*cos((pi*x)/2)-p(x))^2 dx where p=p(x) is a polynomial in P2. Find the point of minimun for E, i.e. find...

Can you tell me is my solution true of the next problem. Find center w_0 and radius R of the circle k, in which the transformation w=\frac{z+2}{z-2} converts the line l:\text{Im} z+\text{Re} z=0. Solution: 2 \to\infty -2i=(2)^*\to w_0...

Hi, I have been told that in R^2 the unit circle {(x,y) | x^2 + y^2 = 1} is smoothly mappable to the curve {(x,y) | x^4 + y^2 = 1}. Can someone please tell me what this smooth map is between them? I can only think of using the map (x,y) --> (sqrt(x), y) if x is non-negative and (sqrt(-x), y)...

Can someone please exPlain to me what the phrase. Which metric do we have to impose in order that the mapping preserves distance means. The example I have is ((-),phi)--->(x,y) = (2a tan(theta/2)cos(phi) , 2a tan(theta/2)sin(phi)) thanks

Hi, does anyone have any idea how to prove the mapping R^2->R^2 (x/(x^2+y^2), y/(x^2+y^2) is a diffeomorphism, and if it is not restrict the values so it is one I am fairly sure it is not over R^2 as it is not continuous at 0, but I don't know what values to restrict it over. I have...

mapping

How does the function f(z) = z + 1/z take a circle of radius g.t. 1 to an ellipse? How do I think about it geometrically ? (i.e., how should I be able to look at the complex function and tell straight away)

I've just started reading up on the holographic principle and eventually want to work my way to figuring out what Verlinde has proposed using it for. One thing I've noticed in a couple of papers is the mapping of the entropy of a black hole onto a holographic screen. Why are families of light...

Hi, everyone: Given a smooth, orientable manifold X, we turn Aut(X) the collection of all self-diffeos. of X into a topological space, by using the compact-open topology. Aut(X) is also a group under composition. The mapping class group M(X) of X is defined as the quotient: M(X):=...

Hi, everyone: I am trying to understand why the mapping class group of the torus T^2 (i.e., the group of orientation-preserving self- diffeomorphisms, up to isotopy) is (iso. to) Gl(2,Z) ( I just realized this is the name of the group of orientation-preserving...

Homework Statement Find a bijective map g : X^{m} \times X^{n} \rightarrow X^{m + n} Homework Equations The Attempt at a Solution I don't even know where to begin. How would I map X^{m} \times X^{n} in the first place? How could I map X^{m + n}?

Hey guys, I'm attempting to map some discrete points on the surface of the Bloch sphere: For instance, the full spectrum of ranges for variable theta is 0 < theta < pi. However, my goal is to limit that range from some theta_1 < theta < theta_2. I was going to use a spherical harmonic...