Read about linear algebra | 393 Discussions | Page 1

  1. J

    I Zero-point energy of the harmonic oscillator

    First time posting in this part of the website, I apologize in advance if my formatting is off. This isn't quite a homework question so much as me trying to reason through the work in a way that quickly makes sense in my head. I am posting in hopes that someone can tell me if my reasoning is...
  2. K

    Linear algebra inner products, self adjoint operator,unitary operation

    b) c and d): In c) I say that ##L_h## is only self adjoint if the imaginary part of h is 0, is this correct? e) Here I could only come up with eigenvalues when h is some constant say C, then C is an eigenvalue. But I' can't find two. Otherwise does b-d above look correct? Thanks in advance!
  3. F

    I Proving linear independence of two functions in a vector space

    Hello, I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
  4. K

    Show that V is an internal direct sum of the eigenspaces

    I was in an earlier problem tasked to do the same but when V = ##M_{2,2}(\mathbb R)##. Then i represented each matrix in V as a vector ##(a_{11}, a_{12}, a_{21}, a_{22})## and the operation ##L(A)## could be represented as ##L(A) = (a_{11}, a_{21}, a_{12}, a_{22})##. This method doesn't really...
  5. K

    What can we say about the eigenvalues if ##L^2=I##?

    This was a problem that came up in my linear algebra course so I assume the operation L is linear. Or maybe that could be derived from given information. I don't know how though. I don't quite understand how L could be represented by anything except a scalar multiplication if L...
  6. K

    I Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices

    I have this problem in my book: Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros. In the picture below is the given solution for this: I...
  7. K

    I Finite fields, irreducible polynomial and minimal polynomial theorem

    I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...
  8. K

    I Calculating A^2 where k = Z_2

    In my book no explanation for this concept is given and i can't find anything about it when I am searching. One example that was given was: Let $$A=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$ with ##k=\mathbb{Z}_2## I think k is the set of scalars for a vector that can be multiplied with...
  9. K

    I Trying to get a better understanding of the quotient V/U in linear algebra

    Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
  10. K

    Linear algebra, find a basis for the quotient space

    Let V = C[x] be the vector space of all polynomials in x with complex coefficients and let ##W = \{p(x) ∈ V: p (1) = p (−1) = 0\}##. Determine a basis for V/W The solution of this problem that i found did the following: Why do they choose the basis to be {1+W, x + W} at the end? I mean since...
  11. J

    I Properties of a unitary matrix

    So let's say that we have som unitary matrix, ##S##. Let that unitary matrix be the scattering matrix in quantum mechanics or the "S-matrix". Now we all know that it can be defined in the following way: $$\psi(x) = Ae^{ipx} + Be^{-ipx}, x<<0$$ and $$ \psi(x) = Ce^{ipx} + De^{-ipx}$$. Now, A and...
  12. MexChemE

    My first proof ever - Linear algebra

    First, a little context. It's been a while since I last posted here. I am a chemical engineer who is currently preparing for grad school, and I've been reviewing linear algebra and multivariable calculus for the last couple of months. I have always been succesful at math (at least in the...
  13. Demandish

    Introductory Linear Algebra Texts

    I am currently enrolled in Multivariate Calculus and am looking to get build up a solid base of mathematics for undergraduate physics curriculum. I am looking for a Linear algebra book that will aid me in my quest. I currently own Axler's Linear Algebra Done Right, but I fear it is too...
  14. S

    Setting Free variables when finding eigenvectors

    upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations. So I took b as a free variable to solve the equation int he following way. But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...
  15. S

    A simple Linear Algebra question: Find all solutions X3 of this system, such that X3 ≠ X1 and X3 ≠ X2

    This is just a small part of a question I have in my assignment and I'm not sure how to solve it, nothing in my eBook or our presentation slides hints at a similar problem, what I tried was I noticed that X1 and X2 have the difference of (3,3,3) and I assume either X3 = (3,3,3) or X3 = (7,8,9)...
  16. S

    Matrix concept Questions (invertibility, det, linear dependence, span)

    I have a trouble showing proofs for matrix problems. I would like to know how A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix holds for square matrix A. It would be great if you can show how one leads to another with examples! :) Thanks for helping...
  17. G

    Fast pentadiagonal matrix solver

    Hello, I'm currently writing a numerical simulation code for solving 2D steatdy-state heat conduction problems (diffusion equation). After reading and following these two book references (Numerical Heat Transfer and Fluid Flow from Patankar and And Introduction to Computational Fluid Dynamics...
  18. B

    Question about linear transformations

    Summary:: linear transformations Hello everyone, firstly sorry about my English, I'm from Brazil. Secondly I want to ask you some help in solving a question about linear transformations. Here is the question: Consider the linear transformation described by the matrix \mathsf{A} \in \Re...
  19. Lauren1234

    Proving Spectrums

    This is my solution so far however I’m not sure where to go from here I think it’s something to do with the trace of the matrix but. This is the full solution but I did row reduction on the matrix K6- $lambda$I
  20. Lauren1234

    Non-abelian groups

    I know for a group to be abelian a*b=b*a I tried multiplying the matrix by itself also but I’m not sure what I’m looking for. picture is below of the matrix
  21. Lauren1234

    I Linear isomorphisms

    how would I go about answering the above question I need some pointers on how to start?
  22. Lauren1234

    I Linear transformations

    Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A). Define T:V→V,f↦df/dx. How do I prove that T is a linear transformation? (I can do this with numbers but the trig is throwing me).
  23. JD_PM

    Matrix representation of a linear mapping

    I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##. This can be represented by a diagram: Thus note...
  24. S

    Find the sampling matrix and sampling structure for R, G and B components

    Hello, everyone. :) All I could gather is that, if I'm correct, lattices are spans of the column vectors of the matrix within the "LAT()" notation and the X and Y occurrences are unit placeholders (such as the pixel unit (since this is in the context of image processing)). And, as an attempt...
  25. A

    I Meaning of each member being a unit vector

    Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged. Hello! I am struggling with understanding the meaning of "each member is a unit vector": I can see that N would represent the number of samples, and the pointy bracket represents an...
  26. christang_1023

    Decompose Involutory Matrix into Difference of Two Idempotents

    I feel confused about proving the two terms are idempotents.
  27. T

    Simplifying a matrix algebra equation (revised)

    I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...
  28. S

    Basis of eigenvectors

    Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} ##. Not sure what to do next. Thanks!!!
  29. F

    Python Invert a matrix from a 4D array : equivalence or difference with indexes

    I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...
  30. E

    Expectation value of operators and squeezing in the even cat state

    I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?