What is the problem with over-determined systems in linear algebra? Do they always have no solution? Is there a proof of that?

erwachsenc6
2022-10-13
Answered

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Steinherrjm

Answered 2022-10-14
Author has **12** answers

An overdetermined system (more equations than unknowns) is not necessarily a system with no solution. If one or more of the equations in the system (or one or more rows of its corresponding coefficient matrix) is/are (a) linear combination of the other equations, so the such a system might or might not be inconsistent.

And some systems which are not overdetermined (number of equations = number of unknowns) have no solutions.

What is true is that whenever we have an inconsistent system of equations, there is no solution.

My point is that you understand that a system of linear equations has no solution if and only if it is inconsistent.

asked 2022-10-16

(6,4); 2x+3y=18

asked 2022-07-09

Suppose we define a summation graph $G$ as follows:

Each vertex $v\in G$ has a unique but unknown value ascribed to it. Each edge $e\in G$ is labelled with the sum of the values of the two vertices it joins.

This construction corresponds to a system of equations where each equation is of the form ${v}_{a}+{v}_{b}={e}_{ab}$ where ${v}_{1}$ and ${v}_{2}$ correspond to the unknown values of two vertices $\u0444$ and $b$, and ${e}_{ab}$ the value of the edge joining the two.

Now, if there is any odd cycle, it's known that there will always be a unique solution for just that cycle. But with an even cycle, there are either an infinite number of possible values, or an inherent contradiction in the values of the edges that make a solution impossible.

So my question is this: is it possible that there is some bipartite graph (which has no odd cycles, but can have a bunch of even cycles joined together) that has a configuration of values with a unique solution?

Each vertex $v\in G$ has a unique but unknown value ascribed to it. Each edge $e\in G$ is labelled with the sum of the values of the two vertices it joins.

This construction corresponds to a system of equations where each equation is of the form ${v}_{a}+{v}_{b}={e}_{ab}$ where ${v}_{1}$ and ${v}_{2}$ correspond to the unknown values of two vertices $\u0444$ and $b$, and ${e}_{ab}$ the value of the edge joining the two.

Now, if there is any odd cycle, it's known that there will always be a unique solution for just that cycle. But with an even cycle, there are either an infinite number of possible values, or an inherent contradiction in the values of the edges that make a solution impossible.

So my question is this: is it possible that there is some bipartite graph (which has no odd cycles, but can have a bunch of even cycles joined together) that has a configuration of values with a unique solution?

asked 2022-07-26

Let $A=\left[\begin{array}{ccc}1& -3& -4\\ -3& 2& 6\\ 5& -1& -8\end{array}\right]\text{and}\overrightarrow{b}=\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\\ {b}_{3}\end{array}\right]$

Show that the equation Ax=b does not have a solution for all possible b, anddescribe the set of all b for which Ax=b does have a solution.

Please explain in steps and especially the finall step towhere you explain for which values b does have a solution and whenthere is none.

Show that the equation Ax=b does not have a solution for all possible b, anddescribe the set of all b for which Ax=b does have a solution.

Please explain in steps and especially the finall step towhere you explain for which values b does have a solution and whenthere is none.

asked 2021-12-13

What is the value of $x-y$ , if $xy=144$ , $x+y=30$ , and $x>y$ ?

asked 2022-05-22

Consider the differential system

$\{\begin{array}{ll}& {y}^{\prime}(t)=ay(t{)}^{3}+bz(t)\\ & {z}^{\prime}(t)=cz(t{)}^{5}-by(t)\end{array}$

with $t>0$

$y(0)={y}_{0},z(0)={z}_{0},\phantom{\rule{1em}{0ex}}a<0,\phantom{\rule{1em}{0ex}}c<0,\phantom{\rule{1em}{0ex}}b\in \mathbb{R}$

the question is to prouve that this system admits a unique solution on $[0,+\mathrm{\infty}]$?

$\{\begin{array}{ll}& {y}^{\prime}(t)=ay(t{)}^{3}+bz(t)\\ & {z}^{\prime}(t)=cz(t{)}^{5}-by(t)\end{array}$

with $t>0$

$y(0)={y}_{0},z(0)={z}_{0},\phantom{\rule{1em}{0ex}}a<0,\phantom{\rule{1em}{0ex}}c<0,\phantom{\rule{1em}{0ex}}b\in \mathbb{R}$

the question is to prouve that this system admits a unique solution on $[0,+\mathrm{\infty}]$?

asked 2022-05-30

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations?

${x}^{\prime}=x+\mathrm{sin}(xy+2)-7$

My idea is to use this fact: Not empty omega limit set - because here we have also bounded functions and omega limit set is invariant. But it's hard to say anything about compactness.

${x}^{\prime}=x+\mathrm{sin}(xy+2)-7$

My idea is to use this fact: Not empty omega limit set - because here we have also bounded functions and omega limit set is invariant. But it's hard to say anything about compactness.

asked 2022-09-03

A collection of 52 dimes and nickels is worth $4.50. How many nickels are there?