UkusakazaL
2020-12-24
Answered

Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero solution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.

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Willie

Answered 2020-12-25
Author has **95** answers

Consider the provided question,

Given a system of five linear equations in six unknowns are all multiples of one nonzero solution.

It can be written as,

We have to explain that the system necessarily have a solution for every possible choice of constants on the right sides of the equations.

Since,A is

Image of A is 5 dimensional subspace of

So,

This means that Ax=b has a solution for every b

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Consider a system of linear equations of the form

$\mathbf{A}\mathbf{x}=\mathbf{b},\mathbf{A}\in {\mathbb{R}}^{L\times K},\mathbf{x}\in {\mathbb{R}}^{L},\mathbf{b}\in {\mathbb{R}}^{K}$

with L variables$x}_{1},{x}_{2},\dots ,{x}_{L}\in \mathbb{R$ and $K\le L$ equations.

We are interested in finding a solution for a single variable x_l. Is there an explicit condition for existence of a unique solution for this variable?

Example: if${x}_{1}+2{x}_{2}+3{x}_{3}=3$ and $2{x}_{2}+3{x}_{3}=2$ , then there exist a unique solution ${x}_{1}=1$ for the variable $x}_{1$ , and we cannot find unique solutions for the other variables.

with L variables

We are interested in finding a solution for a single variable x_l. Is there an explicit condition for existence of a unique solution for this variable?

Example: if