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...
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!
Hello,
I am doing a vector space exercise involving functions using the free linear algebra book from Jim Hefferon (available for free at http://joshua.smcvt.edu/linearalgebra/book.pdf) and I have trouble with the author's solution for problem II.1.24 (a) of page 117, which goes like this ...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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)...
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...
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...
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...
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
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
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).
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...
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...
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...
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...
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!!!
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) ...
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?