rank Definition and 307 Threads

I'm desperately trying to understand how to get from 2.7.11 to 2.7.16 and cannot find any reference online on how to find the inverse of an elastic tangent modulus (fourth_order tensor). Can someone help me or give me a reference I can check where they do a similar thing? I would really...

The published solutions indicate that the nullspace is a plane in R^n. Why isn't the nullspace an n-1 dimensional space within R^n? For example, if I understand things correctly, the 1x2 matrix [1 2] would have a nullspace represented by any linear combination of the vector (-2,1), which...

Let ##M## be a nonzero complex ##n\times n##-matrix. Prove $$\operatorname{rank}M \ge |\operatorname{trace} M|^2/\operatorname{trace}(M^\dagger M)$$ What is a necessary and sufficient condition for equality?

Reference; https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test#:~:text=The Wilcoxon signed-rank test,-sample Student's t-test. I have managed to go through the literature (it is pretty straight forward). In general Wilcoxon-rank method applies to data with unequal variances otherwise...

Find the problem and solution here; I am refreshing on this topic of Correlation. The steps are pretty much clear..my question is on the given formula ##\textbf{R}##. Is it a generally and widely accepted formula or is it some form of improvised formula approach for repeated entries/data? How...

How to write following equation in index notation? $$\nabla \cdot \left( \mathbf{e} : \nabla_{s} \mathbf{u} \right)$$ where ##e## is a third rank tensor, ##u## is a vector, ##\nabla_{s}## is the symmetric part of the gradient operator, : is the double dot product. The way I approached is...

My trouble is being to show A must be of rank 2. Any ideas?

Hi, I was trying to find the rank of following matrix. I formed the following system and it seems like all three columns are linearly independent and hence the rank is 3. But the answer says the rank is '2'. Where am I going wrong? Thanks, in advance!

Hey! :giggle: Question 1: Let $C$ be a $\mathbb{R}$-vector space, $1\leq n\in \mathbb{N}$ and let $\phi_1, \ldots , \phi_n:V\rightarrow V$ be linear maps. I have shown by induction that $\phi_1\circ \ldots \circ \phi_n$ is then also a linear map. I want to show now by induction that if $V$ is...

When I started learning about tensors the tensor rank was drilled into me. "A tensor rank ##\left(m,n\right)## has ##m## up indices and ##n## down indices." So a rank (1,1) tensor is written ##A_\nu^\mu,A_{\ \ \nu}^\mu## or is that ##A_\nu^{\ \ \ \mu}##? Tensor coefficients change when the...

If I have a matrix representing a 2nd order tensor (2 2) and I want to convert this matrix from M$$\textsuperscript{ab}$$ to $$M\textsubscript{b}\textsuperscript{a}$$ what do I do? I'm given the matrix elements for the 2x2 tensor. When applying the metric tensor to this matrix I understand...

I am trying to understand the following: $$ \epsilon^{mni} \epsilon^{pqj} (S^{mq}\delta^{np} - S^{nq}\delta^{mp} + S^{np}\delta^{mq} - S^{mp}\delta^{nq}) = -\epsilon^{mni} \epsilon^{pqj}S^{nq}\delta^{mp} $$ Where S^{ij} are Lorentz algebra elements in the Clifford algebra/gamma matrices...

When discussing how a rank two tensor transforms under SO(3), we say that the tensor can be decomposed into three irreducible parts, the anti-symmetric part, traceless-symmetric part, and a 1-dimensional trace part, which transforms as a scalar. How do we know that the symmetric and...

This should be a trivial question. I am trying to compute the spherical tensor ##T_0^{(0)} = \frac{(U_1 V_{-1} + U_{-1} V_1 - U_0 V_0)}{3}## using the general formula (Sakurai 3.11.27), but what I get is: $$ T_0^{(0)} = \sum_{q_1=-1}^1 \sum_{q_2=-1}^1 \langle 1,1;q_1,q_2|1,1;0,q\rangle...

Homework Statement: Rank in order, from the largest to the smallest, the current densities in these four wires, that have different radius and different electric conductivity Homework Equations: current density resistivity and resistance i guess ja=I/πr2 jb=2I/πr2...

I assumed that the same magnitude of force acts on all sides of the box. Since A had the smallest area, I ranked P(A) as having the largest pressure, followed by P(B) having the second largest and P(top) and P(bottom) having the same pressure at third largest each. However, the ranking I...

Let A be a n x n matrix with complex elements. Prove that the a(k) array, with k ∈ ℕ, where a(k) = rank(A^(k + 1)) - rank(A^k), is monotonically increasing. Thank you! :)

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra ... ... I need some help in order to fully understand the proof of Theorem 12.22 on page 276 ... ...The relevant text reads as follows: In the above...

Hey! :o Let $A$ be a $4\times 5$ matrix with rank $2$ and let $U$ be the corresponding row echelon form matrix. I want to check if the following statements are true or not. If $B$ is a $5\times 5$ invertible matrix, at least two of the columns of $B$ are not in the nulity of $A$. There...

Let $$\left[\begin{array}{rrrrrrr} 1 & 0 & -1 & 0 & 1 & 0 & 3\\ 0 & 1 & 0 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 & 4 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 3 \end{array}\right]$$ Find a basis for the null space of A, the dimension of the null space of A, and...

I am reading Jon Pierre Fortney's book: A Visual Introduction to Differential Forms and Calculus on Manifolds ... and am currently focused on Appendix A : Introduction to Tensors ...I need help to understand some statements/equations by Fortney concerning rank one tensors ... Those remarks...

If we have two sets of coordinates such that x1,x2...xn And y1,y2,...ym And if any yi=f(x1...,xn)(mutually dependent). Then dyi=(∂yi/∂xj)dxj Again dyi/dxk=(∂2yi/∂xk∂xj)dxj+∂yi/∂xk Is it the contravariant derivative of a vector?? Or in general dAi/dxk≠∂Ai/∂xk

Homework Statement Rank the four objects (1kg solid sphere, 1kg hollow sphere, 2kg solid sphere and 1kg hoop) from fastest down the ramp to slowest. (Please see the attached screenshot for more details.) Homework Equations KE_rot = 1/2Iw^2 (where omega = w) The Attempt at a Solution Since we...

Homework Statement Sphere 1 has net positive charge Sphere 2 has net negative charge Sphere 3 has net positive charge The ranking of net charge magnitudes are SPHERE 3 > SPHERE 2 > SPHERE 1 All spheres are conductors Sphere 2 is moved away from Sphere 1 and toward Sphere 3 so that 2 and 3...

Homework Statement Homework Equations None (conceptual) The Attempt at a Solution My logic here is this, Sphere 3 has a net positive charge so it is repelling the positives in sphere 2 and attracting the negatives in sphere 2. This means that D has negative charge and C has positive charge...

Homework Statement Suppose that AB = 0, where A is a 3 x 7 full rank matrix and B is 7 x 53. What is the highest possible rank of matrix B. Homework EquationsThe Attempt at a Solution Since each column of B is in the null space of A, the rank of B is at most 4. I don't understand why it is...

Homework Statement Homework EquationsThe Attempt at a Solution Work = W = ΔKE Case A: ΔKE = WTENSION - WSPRING - WGRAVITY Case B: ΔKE = WGRAVITY - WTENSION Case C: ΔKE - ΔPESPRING = WTENSION - WGRAVITY Case D: ΔKE + ΔPEGRAVITY = -WTENSION Case E: WSYSTEM = 0 Velocity and tension for...

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...

1. Homework Statement In picture Homework Equations f=ma The Attempt at a Solution i can use Newtons equations here, but i am confused about the frictional force.. so it doesn't specify whether it is a frictional or unfrictional surface.. is there something I'm missing here? does friction not...

Homework Statement Two crates, A and B, are in an elevator as shown. The mass of crate A is greater than the mass of crate B. a. The elevator moves downward at constant speed. ... iii. Rank the forces on the crate according to magnitude, from largest to smallest. Explain your reasoning...

Homework Statement How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##? Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...

Homework Statement Rank the rate at which the speed of each object is changing, greatest first. Ignore the sign; rank the magnitude or absolute value only. Homework Equations The red vector is the acceleration[/B] The Attempt at a Solution I have A>F>D>E=C and the program is telling me i am...

Homework Statement Rank the vertical component of the velocity of each projectile at they moment they return to the ground, greatest first. Homework Equations n/a The Attempt at a Solution This is a conceptual homework, so there are no numbers involved. But looking at the picture i was...

Homework Statement I will attatch the picture. Can someone please help me UNDERSTAND how to rank the velocity vectors at the landing zone?? I am having a really hard time with physics >=[ i don't know how to just look at this graph and rank the velocities. Homework EquationsThe Attempt at a...

Homework Statement Rank the vertical component of the initial velocity of each projectile, greatest first.Homework Equations n/a The Attempt at a Solution I took the sine of the angles, the angles are ranked B>A>C>D you can see this visually as well as looking at the maximums. why isn't this...

I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...

I am currently study physics at a top 20 us school with, however, a pretty weak physics department. I am in my sophomore year and almost finished all the major course. Till now, I am still not very into doing REU in my home school, because I feel I should learn more graduate level physics and...

Homework Statement Two computer software packages are being considered for use in the inventory control department of a small manufacturing firm. The firm has selected 12 different computing task that are typical of the kinds of jobs. The results are shown in the table below. At the 0.05 level...

Homework Statement I know that for both method are used to test the 2 group sample for a non-normally distributed population ... But , i am not sure the difference between them . Can someone explain the difference between them ? When to use sign test and wilcoxon signed rank test ? Homework...

Homework Statement Br2, KBr, and HBr Homework Equations none The Attempt at a Solution I understand that Br2 would be least electronegative because they both equally share the electrons, but i don't understand why KBr is more electronegative than HBr. This question was on my quiz, and i...

Greetings, can somebody show me how to calculate such a term? P= X E² where X is a third order tensor and E and P are 3 dimensional vectors. Since the result is supposed to be a vector, the square over E is not meant to be the scalar product. But the tensor product of E with itself yields a...

It is the demonstration of an important theorem I do not succeed in understanding. "A matrix has rank k if - and only if - it has k rows - and k columns - linearly independent, whilst each one of the remaining rows - and columns - is a linear combination of the k preceding ones". Let's suppose...

Homework Statement http://imgur.com/PUrHBaa Question 13 in the middle of the page. Each case in the figure shows an example of force vectors exerted on an object. These forces are all of the same magnitude F_o. Assume the forces lie in the plane of the paper. Rank the cases from greatest to...

Homework Statement hello, I have this graph and i have to figure out these so to speak characteristics. 1) find incidence matrix 2) arrange peak according to rank and layers 3)draw new arranged graph 4) find new connection matricies of peaks and arcs Homework EquationsThe Attempt at a...

Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...

Homework Statement Prove that if rank(A) = 0, then A = 0. Homework EquationsThe Attempt at a Solution This seems like a very easy problem, but I just want to make sure I am not missing anything. rank(A) = dim(Im(A)) = 0, so Im(A) = {0}. Thus, A is by definition the zero matrix. My only...

Homework Statement There is a figure, I'll try my best to draw/describe. 1. All three resisters are in parallel ___R____ !___R____! !___R____ ! 2. 2 resisters are parallel and one in series, after the parallel (ignore the dots) ___R___ ... _____R___ !___R___ ! 3. 2...

Homework Statement T((x_0, x_1, x_2)) = (0, x_0, x_1, x_2) Homework Equations None The Attempt at a Solution I'm getting hung up on definitions. My book says that T is an is isomorphism if T is linear and invertible. But it goes on to say that for T of finite dimension, T can only be an...

Hey! :o Let $\mathbb{K}$ be a fiels and $A\in \mathbb{K}^{p\times q}$ and $B\in \mathbb{K}^{q\times r}$. I want to show that $\text{Rank}(AB)\leq \text{Rank}(A)$ and $\text{Rank}(AB)\leq \text{Rank}(B)$. We have that every column of $AB$ is a linear combination of the columns of $A$, or not...