# What is Echelon: Definition and 50 Discussions

ECHELON, originally a secret government code name, is a surveillance program (signals intelligence/SIGINT collection and analysis network) operated by the United States with the aid of four other signatory states to the UKUSA Security Agreement: Australia, Canada, New Zealand, and the United Kingdom, also known as the Five Eyes.Created in the late 1960s to monitor the military and diplomatic communications of the Soviet Union and its Eastern Bloc allies during the Cold War, the ECHELON project became formally established in 1971.By the end of the 20th century, the system referred to as "ECHELON" had evolved beyond its military and diplomatic origins into "a global system for the interception of private and commercial communications" (mass surveillance and industrial espionage).

View More On Wikipedia.org
1. ### MHB Row echelon form : Can the first column contain only zeros?

Hey! :giggle: Let \begin{equation*}A=\begin{pmatrix}0 & -2 & 2 & 0 & 0 & -6 \\ 0 & 0 & 0 & 1 & 1 & 3\\ 0 & 0 & 0 & 1 & -1 & -1 \\ 0 & 1 & -1 & 0 & 2 & 7\\ 0 & 3 & -3 & 1 & 2 & 14\end{pmatrix}, \ b_1=\begin{pmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0\end{pmatrix} , \ b_2=\begin{pmatrix}-2 \\ 1 \\ -1 \\ 3 \\...
2. ### System of equations and solving for an unknown

The first thing I do is making the argumented matrix: Then I try to rearrange to make the row echelon form. But maybe that's what confusses me the most. I have tried different ways of doing it, for example changing the order of the equations. I always end up with ##k+number## expression in...
3. ### MHB Linear system of equations: Echelon form/Solutions

Hey! 😊 I am looking at the following exercise but I think that I miss something. The statement is the following: We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*} 1) Is the system in echelon form? Justify. 2)...
4. ### MHB Determine the solution set of the system using the echelon form

Hey! :o Let $\displaystyle{a:=\begin{pmatrix}2 & 1 & 0 & 5 \\ 1 & 0 & 1 & 1 \\ 4 & 1 &2 & 7\end{pmatrix}\in \mathbb{R}^{3\times4}}$ and $\displaystyle{b_1:=\begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} , \ b_2:=\begin{pmatrix}-2 \\ 1 \\ 0\end{pmatrix} \in \mathbb{R}^3}$. I applied the Gauss...
5. ### B Finding Bases for Row and Column Spaces

I'm doing problems on finding row and column spaces. My textbook tells me to find the echelon form of the matrix, and then to identify the bases. My question is, can I reduce the matrix to reduced echelon form to get the bases? I have the same question about bases for the solution space.
6. ### MHB Row reduced echelon form and its meaning

Hey. I have the following question to solve: * Given a matrix A that is size m x n and m>n. Let R be the RREF that we get by Gaussian elimination of A. Prove that the system equation Ax=0 has only one solution iff in every column of R there is a leading element. I have some answer of...
7. ### MHB Determine a matrix C such that T = CA has echelon form

Hey! :o Let $$A=\begin{pmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{pmatrix}\in \mathbb{R}^{3\times 3}$$ I want to determine a matrix $C\in GL_3(\mathbb{R})$ such that $T:=C\cdot A$ has echelon form. Performing an elementary row operation is equivalent to multiplying an invertible matrix...
8. ### I Are three zeros always required in the third row for a matrix in echelon form?

I need to find the echelon form of: 1 1 2 8 -1 -2 3 1 3 -7 4 10 and so far I have: 1 1 2 8 0 10 -50 -90 0 0 -52 -104 I was just wondering if I was required to put another zero in my third row. Am I always required to have three zeros in the third row? I'm assuming I do, but when I looked at...
9. ### MHB Find determinant by row reduction in echelon

$\textsf{a. Find the determinants by row reduction in echelon form.}$ $$\left| \begin{array}{rrr} 1&5&-6\\ -1&-4&4 \\ -2&-7 & 9 \end{array} \right|$$ ok i multiplied $r_1$ by 1 and added it to $r_2$ to get $$\left| \begin{array}{rrr} 1&5&-6\\ 0&1&-2 \\ -2&-7 & 9 \end{array} \right|$$...
10. M

### Reducing a matrix to echelon form

Homework Statement (i) Reduce the system to echelon form C|d (ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent. (iii) Repeat part (b) above for k = −18 Homework Equations Gaussian elimination I used here...
11. ### I Can RREF be used for any size matrix?

I understand that the RREF algorithm can be used on matrices representing systems of equations to find the solution set of that system. However, can this algorithm be used for any matrix of any size? For example, what if we, what if we had a 1x1 mattix, or a 2x1? What is the minimum size of a...
12. ### Infinite solution to system with no free variables?

Homework Statement The assignment is to find all values of k (in R) for which the system has 0 solutions, 1 solution and infinite solutions. If there are infinite solutions, find the amount of free variables. The system of linear equations: kx + (k+1)y + z = 0 kx + y + (k+1)z = 0 2kx + y + z =...
13. ### Solve Echelon Matrices Homework with Row Operations & Gaussian Elimination

Homework Statement Find the general solution Homework Equations Row operations Gaussian elimination The Attempt at a Solution This is has happened twice now and I'm not too sure how to deal with it. The last row ends up being all zeros except in that spot. I need to make this into an...
14. ### Reduced Echelon Form - Which is correct?

Hi everyone, I am teaching myself Linear Algebra and I am confused with the terminology used in the subject. I am studying Linear Algebra based on Anton's. In the textbook, an augmented matrix in REF needs to have the first nonzero number in a given row to be 1. However, in other textbooks...
15. ### Is this the correct way to compute the row echelon form?

This is actually a pretty simple thing, but the ref(A) that I compute on paper is different from the ref(A) that my TI-89 gives me. Compute ref(A) where A = \begin{bmatrix} 1 & 2\\ 3 & 8 \end{bmatrix} \\ \begin{bmatrix}1 & 2\\ 3 & 8\end{bmatrix} \ r_2 \rightarrow r_2 - 3 \times r_1 \\ \\...
16. ### System of Linear Equations to Reduced Echelon Form

I have x1 + 3x2 - x3 = b1 x1 - x2 + 3x3 = b1 -2x1 - 5x2 - x3 = b1 So using an augmented matrix I get this [1 3 -1 | 1] [1 -1 3 | 1] [-2 -5 1 | 1] [1 3 -1 | 1] [0 -4 4 | 0] R2 - R1 = R2 [0 1 -1 | 3] R3 + 2R1 = R3 [1 3 -1 | 1] [0 1 -1 | 3] Swap R2 with R3 [0 -4 4 | 0] [1 0 2 | -8] R1 - 3R2 =...
17. ### Proof about system of linear equations in echelon form

Homework Statement Problem: Consider a system of linear equations in echelon form with r equations and n unknowns. Prove the following.: (i) If r = n, then the system has a unique solution. (ii) If r < n, then we can arbitrarily assign values to the n - r free variables and solve uniquely...
18. ### Solving Row Echelon Form: Practicing Tips for Students

For some reason I just can't seem to wrap my head around the idea of reducing a Matrix to row echelon form. I'm familiar with the steps that the textbooks and tutorials use and how it's done but when I try practicing on my own I feel lost. e.g. all I end up with are just a bunch of random...
19. ### Determining Uniqueness of Reduced Echelon Form

Homework Statement Is the reduced echelon form of a matrix unique? Justify your conclusion. Namely, suppose that by performing some row operations (not necessarily following any algorithm) we end up with a reduced echelon matrix. Do we always end up with the same matrix, or can we get different...
20. ### MHB Possible Echelon Form of Matrix

Please check my solution.View image: Possible Echelon Form of Matrix
21. ### MHB Possible Echelon Form of a Matrix

Please Check My Solution View image: Possible Echelon Form of Matrix
22. ### Finding Solutions to Systems of Linear Equations Using Reduced Echelon Form

Homework Statement In each part, the reduced echelon form of the augmented matrix of a system of linear equations is given. Find all solutions to the original system. a.  [1 0] [2] [0 1] [5]  b.  [1 0] [2] [0 0] [1]  c.  [1 0] [2] [0 0] [0]  d. ...
23. ### Linear Algebra: Augmented matrix echelon form y-space?

I'm doing my homework but I'm lost on one thing. Let's say that we have a systems of equations like so: 2x1+3x2=y1 4x1+2x2=y2 Instead of setting it to a constant our teacher sets it to a variable, he says that to be able to compute this, the augmented matrix should look like: 2 3|1 0 4 1|0 1...
24. ### Echelon form and set of solutions.

Hello, I have the following system of linear equations - kx + 3y -z = 1 x + 2y - z = 2 -kx + y + 2z = -1 I have reduced it to 1 2 -1 : 2 0 1 1/4 : 0 0 0 (7-6k)/7k : (8-4k)/7 assuming k ≠ 0. I would now like to be able to determine...
25. ### Reduced row echelon form of a square matrix

I am wondering about the relation betwen RRE forms and identity matrices. Consider the reduced row echelon form of any square matrix. Must this reduced row echelon form of the matrix necessarily be an identity matrix? I would suppose yes, but can this fact be proven? Could anyone provide an...
26. ### Pivot Variables in Reduced Row Echelon Form

Why do we solve for only the pivot variables when we are trying to solve a system of equations using reduced row echelon form? Thank you
27. ### What is the error in this proof of uniquness of row echelon form?

Let's prove the uniqueness of row echelon form (Suppose we already knew existence) First, for any m*n matrix A, think about homogeneous equation AX=0. Obviously, AX=0 has a solution X=0, so its solution set is not empty. And A's row echelon form has same solution set. So if there are more...
28. ### Can you switch rows in a matrix without actually switching them?

Suppose you weren't allowed to switch rows, would it then always be possible to turn a regular matrix into the unit matrix or would the operation be needed in some cases?
29. ### Possible Row Reduced Echelon Forms

This isn't homework. I asked my professor for help on figuring out a way to know the possible combinations of reduced row echelon forms of nxn matrices, or mxn matrices. He only could show me why it was really hard to find this out, not how to actually do it. His method was to use...
30. ### Generic linear algebra echelon form question

Homework Statement What is the reduced echelon form of a n x n nonsingular matrix? Briefly explain.Homework Equations The Attempt at a Solution I know that a n x n nonsingular matrix will always result in echelon form will always have a diagonal orientation with a single digit in its own row...
31. ### Short Reduced Echelon Form Explanation?

Homework Statement If my original echelon form is: 1 1 -2 1 | 2 0 3 3 3 | -3 0 0 0 1 | -4 and according to my notes that my teacher provided, the reduced form is: 1 0 -3 0 | 3 0 1 1 0 | 3 0 0 0 1 | -4 he noted that in the reduced form, the 1's in columns 1, 2, and 4 are...
32. ### Column Space Basis: Why Does Row Reduction Work?

I am a bit puzzled by the following. You know how they teach you that in order to find column space you just need to row reduce the matrix, look at the columns with leading 1's in them and then just read off those columns from the original matrix? Well, why does that actually work? I'm trying to...
33. ### Why is a negative sign not needed when swapping rows in matrix row echelon form?

Hi..I have a very basic query...while solving a determinant, when we exchange/swap 2 rows we need to add a negative sign to the determinant. However, when we are trying to reduce a matrix to a row echelon form, when we swap 2 rows..do we need to add a negatice sign here as well? Well..from what...
34. ### Determinants By Row Reduction/Row Echelon Form

Hello all, I have been studying some linear algebra, and I recently came upon the method of finding determinants by row reduction (to row echelon form). But isn't it true than a matrix can have any row echelon form? If so, this would mean different determinants, right? I am studying from...
35. ### Question about row echelon form

Is row echelon form an upper triangular matrix? if so, does this mean that its determinant could be 1 or 0? Even if its row equivalent has a different determinant? Please Answer and thanks.
36. ### Row Echelon Form: Does Swapping Rows Change Matrix?

Say I am given a matrix and am supposed to put the matrix in row echelon form, does swapping two rows change the final matrix? Say I have: 3 1 4 1 1 1 0 1 3 It would save time to swap the first two rows, however when I do that on my problems the first row is wrong.
37. ### Reduced row echelon form question

My book gives the following definition for reduced row echelon form: 1) If a row has nonzero entries, then the first nonzero entry is 1, called the leading 1 in this row. 2) If a column contains a leading 1, then all other entries in that column are zero. 3) If a row contains a leading 1...
38. ### Why is another row operation necessary to obtain the matrix on the second line?

Homework Statement [PLAIN]http://img260.imageshack.us/img260/727/picture2mg.png just wondering.. isn't the the matrix on the 1st line already in its reduced row echelon form? why is another row operation required to obtain the matrix on the 2nd line? (notice the changes to the matrix on the...
39. ### Reduced echelon form where all variables seem to = 0

Homework Statement Suppose I have the augmented matrix 0 -1 0 | 0 0 -6 3 | 0 0 -1 0 | 0 Homework Equations which equates to -y = 0 and -6y + 3z = 0. The Attempt at a Solution Would the solution be that x, y and z all equal 0? Or do I need to let...
40. ### What is the correct way to convert a matrix to reduced row echelon form?

What exactly is a reduced row echelon matrix. I had to convert this to one: 1 2 1 1 1 1 -3 -6 -2 0 -1 -3 2 4 2 1 3 -3 And got: 1 2 1 1 1 1 0 0 1 3 2 0 0 0 0 1 -1 5 Is this right and, if not, why? Thanks for the...
41. ### Give all the 2x2 row echelon reduced matrices

Homework Statement Give all the possible 2x2 row echelon reduced matrices. 2. The attempt at a solution I thought about the matrices (0 0, 0 0), (1 0, 0 0), (0 1, 0 0), (0 0, 1 0), (1 0, 0 1), (0 1, 0 0), (0 0, 0 1). Where the "," inside the parenthesis means a change of row. So in...
42. ### Compute rank(T) by reducing matrix to echelon form

Homework Statement Let T: R4 --> R4 be defined by T(a,b,c,d) = (4a-b+c, b+3d, 3a+c+d, c-d, b+c+2d) where a,b,c,d are in R. Compute rank(T). Homework Equations The Attempt at a Solution Writing the transformation as a matrix, T[a] = [4 -1 1 0][a] [b] [0 1 0...
43. ### Quick easy question about Row echelon form

So google has yielded no good results. When I "transform" a matrix to row echelon form, not reduced row echelon form (leading entries are not necessarily 1), what does it mean if my last row is all 0's? Another thing, when setting up the equations as a matrix, do I include the solutions of the...
44. ### How to compute row-reduced echelon form and understand upper triangular matrices

I am having problems with understanding the whole concept/how to compute the row-reduced echelon form. Can someone please help me? Thanks
45. ### Row Echelon Form Homework - Finding Mistake

Homework Statement i'm trying to put the 3x3 matrix: [4 2 6] [ 2 8 2] [-1 3 1] into row echelow from. but i don't know where I'm goin wrong in my row operations. could some1 please tell me where i hav...
46. ### Determinant of a matrix using reduced echelon form

Problem statement: Find the determinant of the following matrix by row reduction to echelon form. |1 3 3 -4 | |0 1 2 -5 | |2 5 4 -3 | |-3 -7 -5 2 | I reduced this matrix to |1 3 3 -4 | |0 1 2 -5 | |0 -1 -2 5 | |0 2 4...
47. ### Finding a Different Basis for Echelon Form

Here's the problem: given in echelon form, the column space basis is [5,0,0,0]^t, [4,2,0,0]^t and the question is to find another matrix A with the same echelon form but different basis... how do i find a different basis? thanks
48. ### Reduced row echelon form of matrix

How do you know, when you have to stop row-equivalent operations when you are trying to get a 'reduced row-echelon' form of a given matrix. Is it necessary to have all the columns with pivot element as 1 and rest as 0? do you need to continue the operation if you already have a all 0 row? I want...
49. ### HOw many solutions does this echelon matrix have? Mine isn't right :\

Hello everyone The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have? Here is the matrices: 1 0 -12 0 0 1 0 0 0 0 0 1 0 0 0 0 A. Infinitely many solutions B. No solutions C. Unique solution D. None of the...
50. ### How Do You Solve for x1 and x2 Using Reduced Row Echelon Form?

i have a matrix in reduced row echelon form and it gives me the equations x1+x2=-27.5, x3=-13.5, and x4 = 15. how do i solve for x1 and x2