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. M

    Rank ability of ligands to increase or decrease Δ in a square planar compound

    Homework Statement Homework Equations I am working on a formal lab report for an inorganic chem course and forget how to construct a spectrochemical series. I have 4 complexes, each has a single absorption band in the UV-Vis spectrum. [NiBr(Mes)(PPh3)2] 450nm [Ni(Mes)Br(bipy)]...
  2. A

    Probability of a matrix having full rank

    Hi all, I am trying to find the probability that a matrix has full rank. Consider a K*N matrix where the first K columns are linearly independent columns and the next N-K columns are linear combinations of these K columns. I want to find the probability that a sub matrix formed by...
  3. S

    Question about the properties of the rank of a matrix

    Homework Statement Matrix A is a 4 row by 5 column matrix. Matrix B is a column vector in R^{4}. We are supposed to decide whether the following are (a) no solution, (b) one solution, or (c) infinitely many solutions, or whether (d) the data do not give enough information to tell, or (e) the...
  4. C

    How do i rank magnetic field intensity given amperes?

    Is the magnetic field intensity at point P stronger at 2 amps or 4 amps at point P (just outside each electromagnet), if they both have equal amounts of turns of wire. No relative equations. I know that magnetic field intensity is stronger as the number of loops are increased but I...
  5. A

    MIT Acceptance Despite no first rank

    Did any 1 get into grad school at MIT without being the first ranker at your school? If yes what was it that made you stand out??
  6. matqkks

    Rank of a linear transformation

    Is there a short and simple proof of the Nullity - Rank Theorem which claims that if T: U->V is a linear transformation then rank(T)+Nullity(T)=n where n is the n dimension vector space U.
  7. G

    What is the purpose of introducing the transpose in this proof?

    http://www.viewdocsonline.com/document/8uu6tm You can zoom in on the proof by tabs down to the left. I have understood all of the proof until the last part where they introduce the transpose of A after they have proved that row rank of A is equal or less than column rank of A. Why do...
  8. T

    Finding the rank through row operation

    Is the following statement correct? To find the rank of a matrix, reduce the matrix using elementary row operations to row-echelon form. Count the number of not-all-zero rows and not-all-zero columns. The rank is smaller of those 2 numbers.
  9. A

    Rank of linear mapping

    How come the rank of a matrix is equal to the amount of pivot points in the reduced row echelon form? My book denotes this a trivial point, but unfortunately I don't see it :(
  10. R

    Elementary Linear Algebra - Similar Matrices and Rank

    Homework Statement Suppose matrices A and B are similar. Explain why they have the same rank. Homework Equations The Attempt at a Solution So if A and B are similar, then there is some invertible matrix P such that B = P^-1AP. I have been trying to find some way to relate...
  11. G

    Programs Rank undergrad physics degrees by job potential

    I'm planning to major in physics at my university. However, this will be a second degree. My first was a 3 year biology degree 7 years ago. I'm interested in the job prospects of the various physics degrees and since my university is a small undergraduate school, they offer multiple...
  12. S

    Calculators How to calculate Matrix rank in a Casio fx-9750GA plus

    Hi, I have a graphic calculator Casio fx-9750GA plus, and I'm trying to calculate Rank, I'm not sure that this model have this function and, if it have, I'm not finding it. Anyone knows if it have and how do I get there? Thx
  13. G

    What is the relationship between rank and submatrices in a nonzero matrix?

    Let A be a nonzero matrix of size n. Let a k*k submatrix of A be defined as a matrix obtained by deleting any n-k rows and n-k columns of A. Let m denote the largest integer such that some m*m submatrix has a nonzero determinant. Then rank(A) = k. Conversely suppose that rank(A) = m. There...
  14. G

    Rank of a matrix and its submatrices

    Homework Statement Let A be a nonzero matrix of size n. Let a k*k submatrix of A be defined as a matrix obtained by deleting any n-k rows and n-k columns of A. Let m denote the largest integer such that some m*m submatrix has a nonzero determinant. Prove that rank(A) = k. Now conversely...
  15. S

    Linear Algebra - Rank Theorem

    Homework Statement Suppose a non-homogeneous system, Ax = b, of 3 linear equations in 5 unknowns (3x5 matrix) and 3 free variables, prove there is no solution for any vector b. Homework Equations Using the rank theroem: n = rank A + dim Nul(A) where n = # of columns; dim Nul(A) = # free...
  16. A

    EQUALITY OF ROW AND COLUMN RANK (O'Neil's proof) Is there smt wrong?

    EQUALITY OF ROW AND COLUMN RANK (o'Neil's proof) Is there smt wrong? http://www.mediafire.com/imageview.php?quickkey=znorkrmk3k1otjd&thumb=6 Theorem 7.9: EQUALITY OF ROW AND COLUMN RANK Proof: Page 210. It writes:... so the dimension of this column space is AT MOST r (equal to r if...
  17. L

    Number of poles and rank of controllability matrix

    Hi! I have a little problem. I have an exercise where it's said that the tranfer function gives 3 poles and the rank of the controllability matrix is 4. The question are: how many state has the sistem? Is it controllable? Is it observable? My solution were...the number of state is at...
  18. N

    Is A Full Rank Equivalent to an Overdetermined System?

    Homework Statement Hi If I am dealing with an overdetermined system Ax=b, then I can (assuming A has full rank) find the unique approximative solution by least squares. Now, in my book it says that: "For a full column rank matrix, it is frequently the case that no solution x satisfies...
  19. M

    Prove Matrix Rank of A*A = Rank of A

    Homework Statement Show that for all nxn matricies A with real entries we have rk(A*A) = rk(A) where A* is the transpose of A. Homework Equations The Attempt at a Solution I'm working over a vector space V. Im(A) = {A(v) | vEV} Im(A*A) = {A*(A(v)) | vEV} So Im(A*A) is...
  20. R

    Matrix Multiplication and Rank of Matrix

    Dear Forum, I have one question on matrix multiplication. Suppose there are 2 matrices - A = 1 -1 0 0 2 -1 2 0 -1 B = 1 1 2 and AB = 0 (Zero Matrix) if B not a zero-matrix, then rank(A) is less than s, where s is the dimension of B. I wanted to...
  21. A

    What is the Connection Between Rank and Free Modules?

    I am trying to understand the notions of rank of an R-Module, free-module, basis, etc. I would like to understand this line (expand on it, find some critical examples/counterexamples ,etc) that I am quoting from Dummit & Foote: "If the ring R=F is a field, then any maximal set of...
  22. F

    Some questions about maximum rank & typical rank

    I was wondering some exact explanations of the definition of maximum rank and typical rank-peculiarity of tensors when I read 'Tensor Decomposition and Application' written by Tamara G. Kolda. As his saying, the maximum rank is defined as the largest attainable rank, whereas the typical rank is...
  23. Z

    Rank of Matrix A with Kronecker Symbol and Sum Condition

    helloo while working on a combinatorics problem I have found the following result: let A=(a_{ij})_{1\leq i,j\leq2n+1} where n is a positive integer , be a real Matrix such that : i) a_{ij}^2=1-\delta_{ij} where \delta is the kronecker symbol ii) \forall i \displaystyle{...
  24. A

    Z-score/percentile rank question

    SO the entrance to a college is based off of two factors: 54% grade point average, and 46% supplemental application. Both factors are standardized by the college using a z-score. If a student is ranked in the only ~55 percentile for his grade point average, what percentile rank...
  25. A

    Why Rank is the Trace of a Projection

    Why is the Trace of a projection is its Rank. Thank you
  26. T

    Rank Courses for BS Statistics & Increase Math Understanding

    I am getting my B.S. in statistics in a few years and will then try for a PhD, and I happen to have 1-4 spots where I can take additional courses. I am taking all my stat courses as well as a year of real analysis and a year of abstract algebra and want to take these other courses, but I may...
  27. L

    If an nxn matrix has rank n how do you know it's invertible?

    Homework Statement If an nxn matrix has rank n how do you know it's invertible? The attempt at a solution I know that by the Fundamental Theorem of Invertible Matrices if Rank(A) = n then A is invertible. However, I don't know if that is enough of an answer so it kind of seems like I'm...
  28. quasar987

    Free finitely generated module has finite rank?

    How does one prove that for R commutative, a free finitely generated R-module has finite rank? If R is a field (i.e. in the case of vector space), then we can argue that given a finite generating set S={s1,...,sn}, if S is not linearly independent, then, WLOG, it is that (*)...
  29. B

    Rank of AB: How nxn Matrices A & B Determine Rank

    check that, for any nxn matrices A,B then rank(AB) (> or =) rank A +rank(B)-n
  30. D

    Proving Rank(A)=2 for a 3x3 Matrix with Non-Zero Elements

    Homework Statement Assume that a, b, and c do not equal zero. Let matrix A= 0 a b -a 0 c -b c 0 Prove that Rank(A)=2Homework Equations Definition: The rank of a matrix A is the number of linearly independent rows or columns of AThe Attempt at a Solution I've attempted to get it into reduced...
  31. J

    Setting the record straight on rank, nullity, etc.

    Homework Statement (Pictured) Homework Equations Some Wikipedia and Wolfram MathWorld definitions. In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. The rank...
  32. S

    Understanding Physics with a Rank 4 Tensor Example

    Hi, Can somebody give me an example of a rank 4 tensor in physics? Thanks.
  33. E

    Rank of a Matrix: Why is A = 1 & Not 0?

    Hi I don't understand why only a matrix full of zero has a rank = 0. "the rank of a matrix A is the number of linearly independent rows or columns of A" If I have a 3x3 matrix A = [ 1 1 1 1 1 1 1 1 1 ] assuming a_i denotes the column or row vector i of A. I can say...
  34. P

    Proof: Matrix Rank 1 | 3x3 Matrix A = BC

    Homework Statement 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 Homework Equations rank (BC)=rank (A)=1 rank (BC) \leq rank (B) and rank (BC) \leq rank (C) The Attempt at a Solution I prove that if B is a 3x1...
  35. C

    How to find jordan form given rank?

    How to find jordan form given rank?? Find the jordan for of A given that A is an 8x8 matrix, rank(A)=5, rank(A^2)=2, rank(A^3)=1 and rank(A^4)=0. I know that the largest jordan block will be 4x4 and there will be only one of them since the rank(A^3)=1 but how do i find the rest??
  36. S

    Tensor Rank of 2X2 Matrix: Is It Always 2?

    Should not the definition of "Rank" agree in the two cases below? : 1)rank of a 2X2 matrix and 2) "tensor rank" of the same 2X2 matrix Here is my particular example? |1 1| |0 1| This matrix has rank 2. What is its tensor rank? Still 2? Thnk you
  37. M

    Is this statement about the rank of a linear map true or false?

    Is this statement true or false if false a counterexample is needed if true then an explanation If T : U \rightarrow V is a linear map, then Rank(T) \leq (dim(U) + dim(V ))/2
  38. N

    Mathematica Rank of Matrices in Mathematica

    The problem: I need to find the (minimal) rank of some matrix which is basically all parameters. For example, when i ask for the rank of \begin{pmatrix} a& b& c \\ d& e& f \\ g& h& i \end{pmatrix}, I get 3. I would like to get 1, since (excluding the possibility of a matrix of all 0's) it...
  39. Fredrik

    Finite Rank Operators: Prove T* Has Finite Rank

    This is probably easy. It's really annoying that I don't see how to do this... A finite rank operator (on a Hilbert space) is a bounded (linear) operator such that its range is a finite-dimensional subspace. I want to show that if T has finite rank, than so does T*. I'm thinking that the...
  40. M

    Tensor Rank of Stress Tensors

    I have been trying to learn and visualize a bit of tensor algebra recently, and have been confused by the transformation properties of the stress tensor: Background: *The transformation properties of other tensors have been fairly straightforward for me to grasp so far - one example is the...
  41. L

    Linear Algebra : Rank of a matrix

    Homework Statement Given the following conditions, determine if there are no solutions, a unique solution, or infinite solutions. (Matrix A|B = augmented matrix). Just in case anyone viewing needs a little refresher... Rank = number of non zero rows in the matrix. 1) # of equations ...
  42. S

    Rank the following waves in order

    Homework Statement Rank the waves described by the following equations by their angular frequency, from smallest to largest. A. y(x,t)=(8.0 mm) sin [64 rad/s)t + (8 rad/m)x] B. y(x,t)= (12 mm) sin [(77 rad/s)t + (7 rad/m)x] C. y(x,t)= (16 mm) sin[(16 rad/s)t + (6 rad/m)x] D. y(x,t)= (13...
  43. M

    What is the rank of the matrix of a reimannienne metric ?

    What is the rank of the matrix of a reimannienne metric ?
  44. T

    Rank of a tensor- 2x2x2 Array

    Rank of a tensor--- 2x2x2 Array Can anybody give me an example of 2x2x2 Array whose tensor rank is 2 or Can somebody show me why the tensor rank is two for the following 2x2x2 array. That is can you express as a sum of 2 outer products? I am giving the entries of the first face and then...
  45. M

    Prove that tensor is second rank mixed

    Homework Statement V^alpha and U^beta are both contravariant vectors, and obey the equation V^alpha=E^alpha_beta*U^beta. Show that E^alpha_beta is a mixed second rank tensor. (Note: I couldn't get the latex to work, my apologies for the ugly equations. E^alpha_beta means E with a superscript...
  46. B

    Prove Rank & Similarity of Matrices A & B

    let A and B be n x n matrices over a field F. Suppose that A^2 = A and B^2 = B. Prove that A and B are similar if and only if they have the same rank.
  47. B

    Nullspace, kernel and rank

    Homework Statement for the set of vectors: v_1 = 1, -2, 0, 0, 3 v_2 = 2, -5, -3, -2, 6 v_3 = 0, 5, 15, 10, 0 v_4 = 2, 6, 18, 8, 6 (a) find a basis for the set of vectors and state the dimension of the space spanned by these vectors, what is the rank of this matrix? (b) construct a matrix whose...
  48. B

    Solving the System Ax = b: Is Full Rank Necessary?

    Homework Statement The system Ax=b, with Amxn, and m>n, always has a solution when A has full rank. If False, give a counter example, if True, say why. Homework Equations None The Attempt at a Solution I want to say False because b doesn't need to be in the range of A, so Ax=b...
  49. D

    Proving Rank Relationship Between Matrices A and B

    Homework Statement Prove that for any m x s matrix A and any s x n matrix B it holds that: rank(A) + rank(B) - s is less or equal to: rank(AB) The Attempt at a Solution Obviously, the following are true: - rank(A) is less or equal to s, - rank(B) is less or equal to s, -...
  50. D

    What Determines the Rank and Dimension of a Matrix's Solution Space?

    (a)Determine the row rank of the matrix, 1 1 1 1 1 1 2 5 2 2 0 -6 (b) What is the column rank of this matrix? (c) What is the dimension of the solution space Mx=0 So this is my answer: I have reduced my matrix into echelon form and i...
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