What is rank: Definition and 307 Discussions

Receptor activator of nuclear factor κ B (RANK), also known as TRANCE receptor or TNFRSF11A, is a member of the tumor necrosis factor receptor (TNFR) molecular sub-family. RANK is the receptor for RANK-Ligand (RANKL) and part of the RANK/RANKL/OPG signaling pathway that regulates osteoclast differentiation and activation. It is associated with bone remodeling and repair, immune cell function, lymph node development, thermal regulation, and mammary gland development. Osteoprotegerin (OPG) is a decoy receptor for RANKL, and regulates the stimulation of the RANK signaling pathway by competing for RANKL. The cytoplasmic domain of RANK binds TRAFs 1, 2, 3, 5, and 6 which transmit signals to downstream targets such as NF-κB and JNK.
RANK is constitutively expressed in skeletal muscle, thymus, liver, colon, small intestine, adrenal gland, osteoclast, mammary gland epithelial cells, prostate, vascular cell, and pancreas. Most commonly, activation of NF-κB is mediated by RANKL, but over-expression of RANK alone is sufficient to activate the NF-κB pathway.RANKL (receptor activator for nuclear factor κ B ligand) is found on the surface of stromal cells, osteoblasts, and T cells. Mutations affecting RANK have been associated with infantile malignant osteopetrosis in humans, mice and cats.

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  1. J

    Rank and Weight of a Riemann Curvature Tensor

    Given a Riemann Curvature Tensor. How do you know the weight and rank of each: R^{i}_{jki} R^{i}_{jik} R^{i}_{ijk} Is the Ricci tensor always a zero tensor for diagonal metric tensors?
  2. M

    Proving HomR(F;M) isomorphic to M^n for Free Modules of Rank n < 1

    Let R be a commutative ring with 1. If F is a free module of rank n < 1, then show that HomR(F;M) is isomorphic to M^n, for each R-module M. I was thinking about defining a map Psi : HomR(F;M)--> M^n by psi(f) = (f(e1); f(e2); ... ; f(en)) where F is free on (e1; ... ; en) and show Psi is...
  3. F

    Nullity, rank, image and kernel answer check

    My question is let the linear mapping T : R2->R3 be given by T(x,y)=(x-y,2y-2x,0) write down bases for its image and null-space and determine its rank and nullity. Find the matrix A that represents T with respect to the standard bases of R2 and R3 now i think i know how to do this but I'm...
  4. N

    Relation between rank and number of non-zero eigen values.

    Hi everyone, I've a simple question (the answer may be so trivial that I really ought to be ashamed for asking!) in elementary matrix theory: "Does there exists any relation between the number of non-zero eigen values of a matrix with its rank?" The matrix is taken to be a general (square, of...
  5. Z

    Vector Space, dimensions and kernal rank

    Please could someone help me with this question, thank you. Find dim[Ker(D^2 -D: P_3(F_3) ==>P_3(F_3))] Where dim is dimension, Ker is kernal D is the matrix 0100 0020 0003 0000 D^2 is the derivative of D is it equals 0020 0006 0000 0000 And F_3 is the field subscript3...
  6. WolfOfTheSteps

    Rank of Two Matrices: Is [B AB ... An-1B] = [AB A2B ... AnB]?

    Homework Statement A is nxn B is nxp Is the rank of [B AB ... An-1B] equal to the rank of [AB A2B ... AnB]? If not, under what condition will the ranks be equal? The Attempt at a Solution I don't even know where to start. I know that both matrices should have the same...
  7. A

    Irreducible tensor (second or higher rank)

    When one want to find selection rules for matrix element of (for example) electric quadrupole moment tensor Qij, irreducible components of Qij are needed to apply Wigner-Eckart theorem. When symmetry group is SO(3) irreducible component can be found using what we know from addition of angular...
  8. R

    Proof: Matrix Rank of X = n with Y,Z of Rank n-1,1 Respectively

    I'm trying to show that any matrix X with rank n can be written as the sum of matrices Z and Y with rank n-1 and 1, respectively. Since X,Y, Z have the same dimensions, is this a simple matter of saying pick one of the columns in X with a pivot. Let Z= X with this column replaced by zeroes...
  9. R

    Proving Matrix X rank Decomposition

    How can you prove that matrix X with rank n can be written as the sum of matrices Y and Z where Y has rank n-1 and Z has rank of 1. Thanks!
  10. quasar987

    Rank of a Matrix: Can We Eliminate Lines to Get Non-Vanishing Determinant?

    If A is an nxk matrix of real numbers (n>=k) of rank k, is it true that we can eliminate n-k lines of A to obtain a matrix A' of nonvanishing determinant? I convinced myself of that one time while in the bus and now I can't find the proof.
  11. L

    Help with full rank factorization

    I've been tasked with proving the existence of a full rank factorization for an arbitrary m x n matrix, namely: Let \textit{A} \in \textbf{R}^{m x n} with \textit{rank(A) = r} then there exist matrices \textit{B} \in \textbf{R}^{m x r} and \textit{C} \in \textbf{R}^{r x n} such that \textit{A =...
  12. M

    Finding the rank of a projection of u onto v ?

    hello again, I'm once again stumped, i was asked to find the rank and nullity of the projection u onto v so here is the given: T(u)=ProjvU, where v = <2,4> and this is what i did: let u = <u1 , u2> and plugged everything in the projection formula and ended up with < 4 + 2(u1) , -16 +...
  13. E

    Proving Equal Rank: A Shortcut to Demonstrating Column and Row Rank Equality

    [SOLVED] Proving col rank = row rank Homework Statement Demonstrate these four assertions to get an alternate proof that column rank equals row rank. (a) \vec{y}\cdot\vec{y} = \vec{0} \Leftrightarrow \vec{y} = \vec{0} (b) A\vec{x} = \vec{0} \Leftrightarrow A^TA\vec{x} = \vec{0} (c) \dim...
  14. V

    What is the proof for the similarity of two matrices having the same rank?

    can anyone help me with this proof rank of two similar matrices is same.
  15. S

    Rank of Matrix Problem: Finding k for Rank=2 | Explanation & Solution

    find the value for k for which the matrix A= | 9 -1 11 | |-6 5 -16 | | 3 2 k | has rank= 2 * the spacing on the matrix doesn't seem to want to stay formatted, but it's a 3X3 with row 1= (9, -1, 11), row 2= (-6, 5, -16) and row 3=(3,2, k) The Attempt at a Solution - I...
  16. R

    Rank nullity inequality fom T^2

    Homework Statement Let T be a linear operator on a vector space V (finite dimensional). show that nullity(T^2)<= 2nullity(T) Homework Equations the rank nullity theorem. The Attempt at a Solution i take T to be a lin. transf. from ker(T^2) to V, that is i am restricting the domain...
  17. R

    Rank statistics for jointly normal random vector

    Hi, Just found this forum. It would be fabulous if someone could point me in the direction of a solution to this problem. Consider X, a jointly normal random vector of mean m and positive definite covariance C. I am interested in knowing the probability that the index associated with...
  18. H

    Schools How would you rank these graduate schools?

    Hello. I know that ranking is not everything and it really depends on particular research field and professors. But I do want to get a rough picture on the overall reputation of these graduate schools at this stage. For the field of mechanical engineering or applied mathematics, how would you...
  19. P

    Which fuel is the largest polluter when ranked from 1 to 4?

    The problem asks me to rank from largest polluter to lowest (1 being highest and 4 being lowest.) There are diesel, E85, Gasoline, Gasoline with 20% ethanol. I know diesel will be 1 but then I am kinda confused on which one pollutes more can anyone help me?
  20. F

    Rank of Matrices: Why Equal to Transpose?

    The question in short is, why the rank of a matrix is equal to the rank of its transpose? Matrix is an array of numbers. Then it's amazing to me that the number of linear independent rows coincides with the number of linear independent columns. I tried to find some fundamental answer to this...
  21. C

    Transformation of Rank 2 mixed tensor

    Thanks for the help on the other questions. I am having trouble with another derivation. Unlike the others, it's not abstract whatsoever. Okay I wish to find the transformation Law for the components of a rank 2 tensor. Easy, I know: T: V^* \times V \mapsto \mathbb{R} So T =...
  22. J

    Understanding School Rankings: Debunking the Myth of Purely Positive Research

    How is it define ? Is it only based on positive research results ?
  23. O

    What is the relationship between the rank of a matrix and its transpose?

    Hi, I'm new to the forum but have watched it for some time. I am trying to prove that Rank (A^T) = Rank (A) with A being mxn matrix. I suspect that it has to do with Rank (A) = Row Rank (A) = Column Rank (A) -and- A^T simply being rows / columns transposed but am unsure how to prove. Thanks, John.
  24. R

    Rank of a Matrix and Solving Linear Equations with Vectors

    Homework Statement Find the rank of the matrix A,where A= \left( \begin{array}{cccc} 1 & 1 & 2 & 3\\ 4 & 3 & 5 & 16\\ 6 & 6 & 13 & 13\\ 14 & 12 & 23 & 45 \end{array} \right) Find vectorsx_0ande such that any solution of the equation Ax= \left( \begin{array}{c} 0\\ 2\\...
  25. K

    Rank, Dimension, Subsapce, Column Space

    1) True or False? If true, prove it. If false, prove that it is false or give a counterexample. 1a) If A is m x n, then A and (A^T)(A) have the same rank. 1b) Let A be m x n and X E R^n. If X E null [(A^T)(A)], then AX is in both col(A) and null(A^T). [I believe it's true that AX is in...
  26. R

    Linear Transformation rank and nullity

    Homework Statement Let T: R3 --> R3 be the linear transformation that projects u onto v = (3,0,4) Find the rank and nullity of T Homework Equations So let u=(x,y,z) The Attempt at a Solution So I know that T(u) = proj. u onto v T(u) = [(3x + 4z)/ 25](3,0,4)...
  27. marcus

    Is the popularity of string theory declining as seen through Amazon salesranks?

    Public interest in the big questions is the lifeblood of theoretical science---an important part of what is needed to keep the enterprise vital. From this perspective, public readership of QG books is a matter of concern---particularly books that break out of the mold. In order to track the...
  28. N

    Rank of a Matrix - Explained by an Expert

    Well this is eating my head ! or am I plain stupid ? ... The rank of a Matrix , is determined by the number of independent rows or columns ..Fine .. here's a matrix .. A = 2 4 1 3 -1 -2 1 0 0 0 2 2 3 6 2 5 Apparently " the...
  29. radou

    Exploring the Dimension of Nullspaces in Matrix Products

    Let A and B me matrices such that the product AB is defined. One has to proove that r(AB) <= r(A) and r(AB) <= r(B). My first thoughts are: let A be 'mxn' and B be 'nxp', so AB is 'mxp'. Further on, we know that r(A) <= min{m, n}, r(B) <= min{n, p} and r(AB) <= min{m, p}. I'm stuck here...
  30. T

    Understanding the Rank of a Matrix: Explained Simply

    Hey I am just wondering about this question... I have reduced it as much as I can and the second part of the question is asking about the rank of the matrix... which means the leading number of ones right? SO if I had this matrix 2 5 0 0 2 1...
  31. I

    Linear algebra rank question

    Linear algebra questions (rank, generalized eigenspaces) Hi, This seems to be an easy question on rank, but somehow I can't get it. Let U be a linear operator on a finite-dimensional vector space V. Prove: If rank(U^m)=rank(U^m+1) for some posiive integer m, then rank(U^m)=rank(U^k)...
  32. P

    Prove help. rank of inverse matrix

    I can't find out how to prove this question. Can anyone help? Let A be an n x m matrix of rank m, n>m. Prove that (A^t)A has the same rank m as A. Where A^t = the transpose of A. I seen someone else have asked the question before and had got the answer. However I can't understand it...
  33. Z

    Convert Spearman Rank Test Correlation to P-Value

    I am trying to convert the spearman rank test correlation coefficient to a p-value but I haven't been able to find anything online as to how to go about this. I'm not looking for a calculator, I really would like to know how to convert from this correlation coefficient to a p-value. Any help...
  34. A

    Rank and inverse of matrices

    I have some more linear algebra problems... First: Prove that if B is a 3x1 matrix and C is a 1x3 matrix, then the 3x3 matrix BC has rank at most 1. Conversely, show that if A is any 3x3 matrix having rank 1, then there exist a 3x1 matrix B and a 1x3 matrix C such that A=BC The first...
  35. C

    Rank condition in the Implicit Mapping Theorm

    Hi there. I've recently come across the Implicit Mapping Theorm in my studies and noticed that there is a condition that the rank of the image must be the maximum possible. I'm not directly seeing why this condition is needed, so I was wondering if anyone could provide me with an example of why...
  36. M

    Proof Help - Rank of the transpose of a Matrix

    Hi, I'm having trouble with a proof regarding the rank of the transpose of a matrix. Here's the question: Let A be an m x n matrix of rank r, which is of course less than or equal to min{m,n}. Prove that (A^t)A has the same rank as A. Where A^t = the transpose of A. I can easily...
  37. G

    Linear Algebra and rank problem.

    I have the following problem which I can't figure out. Let A = [a_11,a_12;a_13; a_21; a_22; a_23;] Show that A has rank 2 if and only if one or more of the determinants | a_11,_a_12; a_21,a_22| , |a_11,a_13;a_21,a_23|,|a_12,a_13;a_22,a_23| I know its a 2x3 matrix..which the det...
  38. G

    Question about nullity and rank

    So i have the following matrix: A= [2,0,-1; 4,0,-2;0,0,0] I do r-r-e I get [1,0,-1/2;0,0,0;0,0,0] So my rank for A is 1, because I only have 1 leading one. Now for my nullity, i get the following x_1 - 1/2 X_3 = 0 --> x_1=1/2 x_3 therefore [x_1,X_2,X_3] =[1/2;0;1]t Which would...
  39. H

    Computing rank of a matrix over a finite field

    How would one go about computing the rank of a matrix over a finite field? Obviously row reduction could be used... is there a better way?
  40. J

    A quick question on Spearman's rank correlation coefficient

    Hi, I was just wondering about spearman's rank correlation coefficient hypothesis tests - for these to be valid does the data in the sample have to be drawn from a bivariate normal distribution or does that only apply to the product moment correlation coefficient? Cheers, Just some guy
  41. D

    Understanding Rank and Bases in Linear Algebra: Exploring Column and Row Spaces

    I'm finding it difficult to grasp the concept of rank (more specifically, of bases). First of all, what excactly is a basis? The textbook definition doesn't suffice. What is "column space" (colA) and "row space" (rowA)? If I am given a matrix A and told to find the bases for rowA and...
  42. R

    Linear Algebra: The vector space R and Rank

    Two m x n matrices A and B are called EQUIVALENT (writen A ~e B if there exist invertible matracies U and V (sizes m x m and n x n) such that A = UBV a) prove the following properties of equivalnce i) A ~e A for all m x n matracies A ii) If A ~e B, then B ~e A iii) A ~e B and B~e C, then...
  43. M

    Matrices trouble, finding k so matrix is rank 2

    Hello everyone, I have a problem... I am suppose to Find the value of k for which the matrix: A = -4 9 14 2 7 16 -7 -2 k has rank 2. k = ? I row reduced until i couldn't do it anymore and i got the following: -4 9 14 0 23 46 0 0 92k + 2116 now I'm lost on how I'm suppose to...
  44. M

    I don't understand the rank of a matrix

    Hello everyone, can someone explain to me what the rank of a matrix is? I have the following: 2 3 -2 2 6 0 -4 0 0 Rank = 3; 0 2 0 0 0 0 0 -4 0 0 0 0 9 0 0 0 rank = 3; 1 2 6 -3 Rank = 2; I don't get it! any help would be great!
  45. Y

    Rank of Tensors: Questions & Answers

    q1 What is rank of a tensor? q2 I don't know why after contraction operation (or trace of tensor) the rank of a tensor will be reduced by 2? q3 I can't imagiant how the fourth rank tensor, e^iklm looks like? q4 What does an anti-symmetric tensor e^iklm means? Is it a 4 by 4 martix or a...
  46. M

    Rank magnitudes of the electron's accerlation.

    Hello everyone, I'm lost as usual. Because they are two sheets, infinite and nonconducting. I thought I would use this equation: E = \delta /(2Eo). But they give the separation which i don't see how that fits into this equation. I figured I could find the accerlation using F = MA. But...
  47. K

    Matrix Rank Proof: A ≤ m when Rank A = m

    Q: If A is an m x n matrix and rank A = m, show that m <= n. I know that by definition if A is m x n, then rank A <= m and rank A <= n. However, I do not know how I would do this if rank A = m. Any help would be great thanks.
  48. P

    How Does Matrix Rank Relate to Vector Independence and Dimensionality?

    Suppose that a matrix A is formed by taking n vectors from R^m as its columns. a) if these vectors are linearly independent, what is the rank of A and what is the relationship between m and n? is the rank the same as the dimension of the column space, or n, and m less than or equal to n...
  49. D

    Rank Osmosis Scenarios & Define Osmosis

    Hey guys can someone help me with the osmosis scenarios and rank them in order of the most mass gained and also write a good definition of what osmosis is thanks. thanks first of all i have the following solutions i have 5% sucrose in dialysis tube in distilled water - cup 1 i have 10%...
  50. E

    Verifying the Rank of Matrix A

    This is a T/F - prove type of question: A is m x n, M is matrix of TA with respect to bases B of R^m and B' of R^n. Then rank of A = rank of M. My reasoning is that it is true, since the lin. transf. is R^n->R^m, which means that in this formula: M = CB' A PB (CB' (coord matrix) is...
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