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

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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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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 ...
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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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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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34 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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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 ...
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1answer
81 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 ...
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1answer
20 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 $ ...
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2answers
43 views

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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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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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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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
22 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 ...
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1answer
99 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 ...
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40 views

Fourier transform of a Gaussian

I am trying to solve the following exercize: Show that Fourier transform of a Gaussian (a function of the form $Ae^{-\frac{x^2}{\sigma^2}}$) is also a Gaussian. So I did the required calculation (I ...
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1answer
68 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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34 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
28 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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43 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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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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126 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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27 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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52 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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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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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
99 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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44 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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314 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
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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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23 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 \\ ...
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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
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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: $$ ...
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1answer
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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 ...
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1answer
45 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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42 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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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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59 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 ...
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80 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 ...
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29 views

Lower Triangular Form

Suppose that A is an nxn matrix of real numbers such that each entry satisfies |aij| <= 1. If Gaussian elimination with partial pivoting is used to reduce the matrix to lower triangular form, show ...
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Gaussian Elimination on Block Matrix

I have a large matrix, with possibly over 100,000,000 elements in it and I want to solve it quickly. I want to take advantage of the fact that the matrix is partitioned into a small number of unique ...
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Linear algebra: relationship between vectors and lines/planes/hyperplanes

In my linear algebra class and in every online source I can find, they seem to want to introduce matrices and row reduction as the intersection of lines/planes/hyperplanes (depending on dimension). ...
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Basin of attraction (Eigenvectors)

First of all, the transition matrix is given as: $$ M= \begin{matrix} \frac 45 & 0 & 0 \\ 0 & \frac 65 & 0 \\ 0 & 0 & 1 \\ \end{matrix} $$ ...
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Gaussian elimination involving parameters

The problem is :Solve the given system of equations involving the parameter a : $$x+y+az=1\\ x+ay+z=a\\ ax+y+z=a^2\\ ax+ay+az=a^3 .$$ I tried to solve this using the Gaussian method but I'm stuck ...
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Consider the following system and find the values of b for which the system has a solution

So I have this system: $$\left\{\begin{array}{c} x_1 &− x_2 &+ 2x_3 &= 2 \\ x_1 &+ 2x_2 &− x_3 &= 2 \\ x_1 &+ x_2 & &= 2 \\ x_1 & & +x_3 ...
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1answer
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Method to find the set S of reals $λ$ such as $rg($M-I3)<3 given a matrix

Considering the endomorphism $f$ of $R^3$ of \begin{bmatrix} -3 & 5 & -5\\ -4 & 6 & -5\\ -4& 4 &-3 \end{bmatrix} relatively of the canonical base bc of $R^3$ find the ...