For questions on or related to the technique of Gaussian elimination, used in solving systems of linear equations.

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Gaussian Elimination - contradicting results

I am having trouble with the Gaussian Elimination algorithm, doing it in 2 different ways, but still doing it properly leads to contradicting results, one stating that we have 1 solution and the other ...
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0answers
19 views

Online algorithm for reduced row echelon form

Rough Definition of "Online Algorithm" In computer science an online algorithm is used to calculates a function of a set, but is fed its inputs incrementally instead of at once. As a rule they ...
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1answer
18 views

Computing $PAQ = LU$ using Gaussian elimination with complete pivoting

Suppose $PAQ = LU$ is computed via Gaussian elimination with complete pivoting. Show that there is no element in $e_i^{T}U$ i.e., row $i$ of $U$, whose magnitude is larger than $|\mu_{ii}| = |e_i^{T}U ...
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1answer
48 views

How do I rearrange this matrix equation to find A and b?

The Question: It is possible to rearrange the matrix equation $\pi^TP= \pi^T$ into a linear system $Ax = b$ where $x = \pi$ is the unique solution to the system. Such a system could be solved by, ...
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2answers
48 views

Solving linear equation

I have learned how to solve linear equation with Gauss-Jordan Elimination method, but I have came across a type of equation I don't know how to solve using that method. I tried other methods but didn'...
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1answer
28 views

How to use elimination for five equations?

$y_i = C/x_i ^q$ , i = 5 I am trying to find the average q from different points for y and x but how can i eliminate C I tried elimination but there's 5 equation ,I'm confused can someone help me?
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1answer
27 views

Gaussin Elimination preserves S.P.D.

Let $A \in \mathbb{R}^{n \times n} $ be symmetric positive definite with positive diagonal entries. I'm trying to show that at each step $m$ of gaussian elimination $$ a^{(m+1)}_{ij} = a^{(m)}_{i,j} ...
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1answer
32 views

Why am I not getting two positive pivots for this symmetric matrix?

I'm watching Gilbert Strang's linear algebra lecture on positive definite matrices and I have a question about the example he uses in the last 10-15 minutes of the video. Strang is demonstrating the ...
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0answers
16 views

Property of Gauß Elimination

I have the following problem: Suppose $A\in\mathbb{R}^{m,n}$ has the property that after applying the Gauß Elimination we obtain a row-echelon form but without using changes of rows.The claim should ...
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1answer
51 views

Trouble to obtain eigenvectors of a matrix knowing its eigenvalues

The problem: Being given the matrix: $$ \begin{bmatrix} 0 & -1 & -1 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{bmatrix}$$ and two of its eigenvalues $0,...
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18 views

Question with Gauss-Jordan elimination produces the matrix

Please help me explain the problem below: How can I use Gauss-Jordan to get all bottom roll become 0? Thank you so much!
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2answers
15 views

Describe the region (point, line, or plane), if any, that is the intersection of these planes?

The following equations represent planes in $R^3$: $y-x-3z = 1$ $z+2y-3x=-4$ $3y-4x-2z=-3$ If put into augmented form, you get -1 1 -3 | 1 -3 2 1 | -4 -4 3 -2 | -3 And ...
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2answers
35 views

Determine if the set of vectors are linearly independent or linearly dependent

$$u=(1,1,1,3)$$ $$v=(1,2,1,3)$$ $$w=(1,2,3,2)$$ I need help understanding the method of how to do solve this type of problem. I understand that the concept is just to find out if the constants $k_n$ ...
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0answers
7 views

Replacing a Linearly Dependent Column in a Matrix with a Linearly Independent One to Obtain the Inverse of a Matrix

I am working on an encoder and decoder for a Random Linear Fountain. Basically, say I have 10 bits. In order to encode these, at each clock cycle, labelled by n, the encoder generates K random bits $...
2
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1answer
82 views

Why is Gaussian elimination still being taught when there are more efficient methods? [closed]

Why is Gaussian elimination still being taught when there are more efficient methods? By efficient methods I mean less time and effort consuming ways to solve a system of linear equations. I have ...
2
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1answer
22 views

Block Matrix Nonsingular $\iff v^T A^{-1} u\neq 0$

Here is the given question and my work so far: Question: Let $A$ be an $n \times n$ invertible matrix, and let $u$ and $v$ be two vectors in $\mathbb{R}^n$. Find the necessary and sufficient ...
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4answers
38 views

Are these vectors also a basis of the subspace?

So I did a Linear Algebra test today and had to decide a basis for the subspace $ [(1,2,3,2),(0,1,8,5),(-2,-4,-6,-4)]\;in\;\Bbb R^4 $. The correct answer acording to the solution was $ \{(1,2,3,2),(0,...
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2answers
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Quick Eigenvalues Question

I'm asked to find the eigenvalues and bases of the eigenbases for this 3x3 matrix. $$\begin{pmatrix} 1 &0 & -2\\ 0 &0 & 0 \\ -2 &0 & 4\end{pmatrix}$$ ...
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31 views

Elimination Method - 4 variables?

$x+2y - 11z = 0$ $-5y-15z=0$ $3y+9z=0$ $x+y-14z = 0$ I got $y=-3z$ and $x=17z$ , but I can't go further. Any help would be greatly appreciated. Thanks.
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0answers
21 views

Gaussian process regression from predictions

Can any body provide example of a gaussian process regression being used to generate confidence intervals?
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0answers
15 views

Finding all rows with 2 nonzero columns in a matrix

[moved from stackexchange] For this purpose it doesn't matter what type the values of the matrix are, so let's assume R. I want to find all rows that have 2 (or less) nonzero columns, including if ...
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2answers
25 views

Explanation of backwards substitution in Gaussian elimination

I'm not sure what the back subsitution is doing on Gaussian Elimination... I understand how it is trying to get the upper triangular matrix with the 0s under the diagonal, and so I get the why we'...
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1answer
106 views

Euclid's algorithm and Gaussian elimination: most computationally efficient approach

I think this is more a mathematics question than a computer science one - but as it is about computational complexity it could be either, so apologies if you think it is in the wrong place... The ...
2
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1answer
74 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
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48 views

Solving a large set of equations with Newton Raphson and Gauss Jordan Elimination in VBA Excel (6x6)

What I tried to do is to solve a system of 6 equations with VBA excel. This system counts some nonlinear equations which requires Newton Raphsons method to solve and Gauss Jordan Elimination to invert ...
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1answer
33 views

Given a set of (0,1)-vectors, find a subset which adds to the zero vector mod 2

I am reading Quadratic sieve in wiki and it present the next problem: Given a set of (0,1)-vectors, find a subset which adds to the zero vector mod 2 My question is simple: How to solve it? ...
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1answer
22 views

Finding solutions using Gauß-Jordan-Algorithm

I have matrix A = \begin{bmatrix}1&2&3&4&5\\2&4&3&5&4\\3&6&5&8&7\end{bmatrix} and\vec b = \begin{bmatrix}1\\2\\3\end{bmatrix} I expanded the matrix ...
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1answer
44 views

Solving matrix using Gaussian elimination and a parameter

$\begin{bmatrix} x_{1} & 2x_{2} & & & ax_{5} & x_{6} & = & -2 \\ -x_{1} & -2x_{2} & & & (-1-a)x_{5} ...
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2answers
114 views

how to find null space basis directly by matrix calculation

The problem of finding the basis for the null space of an $m \times n$ matrix $A$ is a well-known problem of linear algebra. We solve $Ax=0$ by Gaussian elimination. Either the solution is unique and $...
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1answer
243 views

Minimize the sum of solution of linear equation

Let x(i,j) be a variable. All variables and constants can only have value of 0 or 1. Also, sum of two variables x(i,j) and x(k,l) is equal to (x(i,j)+x(k,l)) % 2 ...
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31 views

Unexpected rank of a matrix

I have a sparse matrix with 100 rows and, when I do a Gauss decomposition, I get a matrix with 90 rows. But if I remove tha last row of the first matrix, now with 99 rows, and I do a Gauss ...
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59 views

How to solve linear system of equation over a finite field

How can we solve linear system of equation over a finite field? I just found out about finite fields and i am having a hard time understanding solutions given on net. I know its equivalent to ...
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2answers
55 views

How do I choose a free variable?

I have a question regarding the Gaussian method for solving linear equations. I had to solve 2 equations with 3 unknowns and naturally with the elimination process I had 2 variables left. I thought ...
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2answers
50 views

How can we solve the linear system? [closed]

How can we solve the following linear system ? $$ax+by=0 \\ cx+dy\neq 0$$
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1answer
24 views

Gaussian elimination in linear equations

So I have problems with solving this question, can someone please help me out with this one? I really don't need a direct solution I just need to know how should I do it, or if u are generous enough ...
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1answer
109 views

Complete Pivoting VS Partial Pivoting in Gauss Elimination

I have a hard time understanding that when and under what conditions we can use Gauss elimination with complete pivoting, and when with partial pivoting, and when with no pivoting? (I mean what is the ...
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46 views

What does the size of a matrix tell about its dimensions?

I am having trouble understanding a lot of terminology here. When inspecting a matrix columns and rows, what does it indicate -- dimension of the vectors maybe, or anything else? Following that, what ...
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4answers
330 views

What is an intuitive explanation to why elimination retains solution to the original system of equations?

I've studied linear algebra before, however, I wanted to come back to the foundations and understand it again from the beginning. I was looking the following inoffensive linear equations: $$ x - 2y =...
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2answers
51 views

Explanation of Gauss-Jordan elimination method.

I know how to solve the system of linear equations, how to find inverse of matrix etc. by the Gauss-Jordan method. But I want to understand why this method works (in cases of inverse matrix ...
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1answer
24 views

Basic Gaussian elimination

I just finished a course about systems of linear équations and I'm trying to do some exercises. Here is the system to solve using Gaussian elimination: $$ \begin{cases} -5x-2y+z=a \\ -4x-4y-4z=b \\ x-...
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2answers
93 views

Inverting a matrix in $\mathbb{Z}/n\mathbb{Z}$.

So in my Linear Algebra course I was shown that we cannot directly use row reduction to invert a matrix over a commutative ring in general because the algorithm requires elements to be invertible (...
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2answers
100 views

Solving a system of linear equations with parameters

I've been given the following system of equations: $$ (4-\lambda)x_1-2x_2-x_3=1\\ -2x_1+(1-\lambda)x_2+2x_3=2\\ -x_1+2x_2+(4-\lambda)x_3=1 $$ The resulting coefficient matrix would be: $$ \begin{...
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1answer
73 views

In Gauss-Jordan elimination, what does it mean to 'restore' a row?

As seen here I am familiar with the elementary row operations but I've never heard an step worded in this way. Thanks!
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1answer
52 views

Determine if the linear systems $A \vec x = \vec 0$ and $B \vec x = \vec 0$ are equivalent.

For one of my homework assignments, the question posed is as follows: Determine if the linear systems $A \vec x = \vec 0$ and $B \vec x = \vec 0$ are equivalent where: see matrices To solve, I ...
2
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1answer
51 views

Row & Column Operation to Determine Rank

While evaluating the rank of a matrix is it permissible to apply row and column operations simultaneously on a single matrix? Most of the books that I discussed use either row or column operation (but ...
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1answer
50 views

Matrix question, all zeros with constant

What does it mean if the bottom row of a matrix is all zeros followed by a constant? Example Solve the given system of equations using either Gaussian or Gauss-Jordan elimination. x - y + z = 0 -x + ...
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40 views

Show that n(n+1)/2 multiplications are required

$a_{11}x_1$+$a_{12}x_2$+$a_{13}x_3$+ ...+ $a_{1,n-1}x_{n-1}$+$a_{1n}x_n$ =$b_1$ $a_{22}x_2$+$a_{23}x_3$+ ...+ $a_{2,n-1}x_{n-1}$+$a_{2n}x_n$ =$b_2$ $a_{33}x_3$+ ...+ $a_{3,n-1}x_{n-1}$+$a_{3n}x_n$ =$...
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0answers
61 views

Finding matrix inverse using Gauss method

I have been trying to find the inverse of a matrix using Gauss method and I want to know suppose what happens if I don't get the "1" in reduced matrix on the left? Does it mean that the inverse doesn'...
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0answers
112 views

numerical stability: LU decomposition

I'm trying to evaluate the numerical stability of LU decomposition. I implemented code in java to calculate the inverse matrix with LU. I made 3 attemps. a) mantissa 4 b) mantissa 6 c) maschine ...
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1answer
37 views

Use Gaussian elimination to convert matrix A to row echelon form R.

Can someone please help me with this question if you can? I have done the ERO's, but I did 5 instead of the 3 that it is asking for and I cannot seem to get it down to 3. I'm not sure if I am reading ...