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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?