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ChemistryNat

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## Homework Statement

SHow that the set of solutions to a homogenous system of m linear equations in n variabes is a subspace of [itex]ℝ^{n}[/itex] (Show that this set satisfies the definition of a subspace)

## Homework Equations

## The Attempt at a Solution

If {V1,...Vk}=[itex]ℝ^{n}[/itex] then every vector [itex]\vec{q}[/itex][itex]\in[/itex]ℝ can be written as a linear combination of the set

c1V1+....+ckVk=[itex]\vec{q}[/itex]

This system of linear equations must have a solution for every [itex]\vec{q}[/itex][itex]\in[/itex]ℝ and therefore the rank of the coefficient matrix = n

If the rank of the coefficient matrix of a system

c1V1+...+ckVk=v

is n, then the system is consistent for all V[itex]\in[/itex]ℝ

∴ {V1,....,Vk}=[itex]ℝ^{n}[/itex]

I thought I was on the right track, but a theorem in my textbook says

" Let [A|[itex]\vec{b}[/itex]] be a system of m linear equations in n variables. Then [A|[itex]\vec{b}[/itex]] is consistent for all [itex]\vec{b}[/itex]=[itex]ℝ^{n}[/itex] if and only if rank(A)=m"

Does the requirement change is they are homogenous? Am I even on the right track?