rank Definition and 290 Threads

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 :(

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...

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...

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

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...

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...

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...

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...

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...

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...

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...

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...

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{...

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...

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

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...

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...

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 (*)...

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

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...

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...

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

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...

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...

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??

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

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

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...

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...

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 ...

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...

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

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...

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...

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.

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...

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...

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, -...

(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...

Homework Statement Prove Rank A + dim Nul A^T = m where A is in R^(mxn) Homework Equations The Attempt at a Solution I honestly can't figure out where to go with this. I know that Rank A + dim Nul A = n, but I don't know if there is a relationship between the two.

Let S = {v1, v2, v3, v4, v5} v1 = <1,1,2,1> v2 = <1,0,-3,1> v3 = <0,1,1,2> v4 = <0,0,1,1> v5 = <1,0,0,1> Find a basis for the subspace V = span S of R^4. ---- My attempt: I place the five vectors into a matrix, where each vector is a row of the matrix. I solve for row-echelon (not RREF). I...

Hi,all I am really very serious about what actually is the nature of pressure as a physical quantity.Books says it has no direction i.e. it is scalar some says it is not.but thinking ourselves it seems pressure has direction in the direction of applied force.Now I want to understand the...

Hello everybody, yesterday I stand to teach vectors and scalars to 12th standard students in a coaching.While giving examples of scalars I named mass , work , pressure etc.Then a student argued me that pressure should be a vector quantity since when you apply a push on wall that is force then...

Homework Statement Find either the rank or nullity of T. T:P2--> R2 defined by T(p(x)) = [p(0) p(1)] Homework Equations Null(T)={x:T(x)=0} I think its usually easier to to find Nullity as opposed to Rank. The Attempt at a Solution I...

1. Prove that the rank of a matrix is invariant under similarity.Notes so far: Let A, B, P be nxn matrices, and let A and B be similar. That is, there exists an invertible matrix P such that B = P-1AP. I know the following relations so far: rank(P)=rank(P-1)=n ; rank(A) = rank(AT); rank(A) +...

If I measure an angle in one reference frame to be 90 degrees, would it be 90 degrees with respect to all other reference frames? That is, is angle measurement a rank 0 tensor? I'm assuming all other reference systems are at non-relativistic velocities.

"The components of a vector change under a coordinate transformation, but the vector itself does not." ie: V = a*x + b*y = c*x' + d*y' Though the components (and the basis) have changed, V is still = V. Question 1: Is that right? (I'm assuming so, the main Q is below) Tensor rank...

Homework Statement Let A=[{1,3,2,2},{1,1,0,-2},{0,1,1,2}] i) Find the rank ii) Viewing A as a linear map from M4x1 to M3x1, find a basis for the kernel of A and verify directly that these basis vectors are indeed linearly independent. Homework Equations None The Attempt at a Solution...

I have the following problem: A * Phi = Ax' * Sx + Ay' * Sy where, A= Ax' * Ax + Ay' * Ay + Axy' * Axy and I would like to solve for Phi. Matrix A is: 1)symmetric 2) [89x89] 3) Rank(A)=88 ( I guess it means that there is no unique solution ) 4) Det(A)~=0 ( I guess it means...

Prove that the function [ ]B: L(V) -> Mnxn(R) given by T -> [T]B is an isomorphism. [T]B is the B-matrix for T, where T is in the vector space of all linear transformations. I don't quite understand this...