1. H

    Finding the image of a region under a complex mapping

    $$w = 2\left(\frac{(z^3-i)}{(z^3+i)}\right)^{1/4}$$ $$D = \{z\mid 0 \leq \operatorname{Arg}(z) \leq \pi/3\}$$ Find the image of the region $D$ under the mapping $W(z)$. I am a Civil Engineering major and found this problem to be very difficult. A description and plot will help too. I...
  2. E

    Normality of the inverse image can't be proved using global methods?

    Notation: x^g= gxg^{-1}, H^g= gHg^{-1}. "in" means belonging or inclusion depending on the context. Let f:G --> L surjective homomorphism, K normal in L, H= f^{-1}(K). Then H normal in G. Proof: let g in G. Then f(H^g)= f(H)^f(g). But f(H)= f(f^{-1}(K))= K because f onto. Hence f(H^g)= K. Let...
  3. A

    Question regarding the Image of a linear map

    Let $V$ be a vector space of finite dimension, and let $T: V\rightarrow V$ a linear map. Why is it true that there must me a $k$ for which $ImT^k=ImT^{k+1}$? I can see why it's true when $T$ is not invertible, since in this case, $T$ is nilpotent of some order, and this $k$ is that order. But...
  4. A

    Help with proving that kernel is a proper subset of the image of a linear map

    Let $V$ be a vector space of dimension 3, and let $T:V\rightarrow V$ be a nilpotent linear operator of order 3. I need to prove the following: 1. Suppose $v\in V$ is a vector for which $T^2(v)\neq 0$, prove that the set $B=\{v,T(v),T^2(v)\}$ is a basis for $V$. 2. Prove that $Ker(T)\subset...
  5. A

    How to find the Image of this linear transformation?

    Suppose we take our vector space V to be \mathbb{R}^{2\times2}, and let's say T:V\rightarrow V is given by: T\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr)=\bigl(\begin{smallmatrix}a & b-c\\ c-b & a\end{smallmatrix}\bigr) I want to find the kernel and image of the above...
  6. S

    solve IVP using laplace transforms: please check my work in attached image

    Original problem: y'' + 7y' + 10y = 3e^(-2t) - 6e^(-5t) and y(0) =0 and y'(0) = 0
  7. E

    Proof that the inverse of a function is the mirror image of the function? (GR. 11)

    If (a, b) = f, then (b, a)= f^-1. The claim has been made that the graph of the inverse of a function is the mirror image of the function, reflected in the line y = x. To prove this you must a) Show that the line that passes through the points (a, b) and (b, a) intersect the line y = x at a...
  8. D

    Image of transformation

    V and W are vector spaces and there is transformation T:V->W So they defined the image as ImT={Tv | v is V} So does W=ImT?
  9. sakonpure6

    Finding the image point on an inverse Exponential Function.

    There is a point (1,0) of the parent function f(x) = log x find the image point on f(x)= 5 log(x-2) which is the same as y= 5f(x) . I know how to do this by applying appropriate transformations, however how come when I use the the algebraic approach found here...
  10. S

    How to get mirror image of knot

    Dear mathematicians Kindly I have a small question: The projection of a knot in R^2 can be obtained by choosing a vector v in R^3 and 2-dimensional hyperplane in R^3 that is disjoint from the knot such that we project the knot in the direction of v onto the hyperplane. Let -K denote the...
  11. R

    To show that the image of a continuous function defined on a compact set is nonempty.

    Given a continuous function f defined on a compact set A, to show that f(A) is nonempty, the author of my textbook simply says that "every continuous function defined on a compact set reaches a maximum." Can anyone explain it in more detail? Why is that? Or how should I prove its...
  12. A

    Kernel and image of linear operator

    Hi, I'm Kalish. I have a problem and a proposed solution. Please point out any errors! Not sure about this one. Here's my work for it: Problem Statement: Determine the dimensions of the kernel and the image of the linear operator T on the space R^n defined by...
  13. sakonpure6

    Finding Image of a point.

    Nvm, got it. Sorry for the inconvenience.
  14. O

    Calculating resize amount needed to match of image sizes of difference FOVs (AOVs).

    Hi, I am trying to work out the scaling factor required to match up the size of objects in 2 photos, taken from the same position, with different (known) Field/Angle of views. (You could also use it to find out the FOV of one image if you know the FOV of another and work out the scaling factor...
  15. L

    Use Gram-Schmidt process to find an orthogonal basis for the image space A. QR Factor

    Question is attached. I just would like someone to verify whether the answer involves fractions/mixed numbers because I do tend to get antsy when I see them in my work. I wonder if I did it right. So anyway here is what I got with Gram-Schmidt Process: With x1, x2, and x3 representing the...
  16. T

    Compose area formula from image

    Hello everyone, The prelude to an optimalisation problem with areas is the composition of the area formula. However, I have come across an image from which I just cannot seem to create the formula, however I have got the feeling that it should be incredibly easy. Could someone please take a...
  17. J

    image & kernel

    When does the kernel of a function equal the image? Thanks in advance
  18. E

    Star photography, camera image scaling problem

    I want to find out what angle a pixel in the images which my camera produces represent. So I take pictures of two points (stars in the sky). The true angle between them is known beforehand. But the camera produces images where the pixels are rectangular, i.e. non-quadratic. They are more...
  19. W

    Determining an area from an image profile plot

    Dear all I have a graph which is that a of a profile plot. I would like to quantify a certain region of my graph and I want to come up with a formula for it. Any suggestions will help . I am attaching an image and in red is the area i want to come up with an equation for. The graph is...