Aine Sellers

2022-02-22

This question from linear algebra
Suppose you [ have a consistent system of linear equations, with coefficients in R, which are homogeneous - that is, all the ${b}_{i}$ are 0. Explain why the set of solutions to this system forms a vector space over $\mathbb{R}$. Then, explain why if the system was not homogeneous (i.e. if at least one of the ${b}_{i}$ is nonzero) the set of solutions would definitely NOT form a vector space over $\mathbb{R}$.
Who knows?

Haiden Frazier

It is given that we have a consistent system of linear equations with coefficients in $\mathbb{R}$. So, the system is written as Ax=0 which is a homogeneous system with all the bi equal to 0 when Ax=b.
We know that the system Ax=0 is said to be consistent if it has a solution. We have to show that the set of solutions of this homogeneous system form a vector space over $\mathbb{R}$.
Property 1: The set of solutions contains the zero vector.
If we take the vector x as a zero vector i.e., x=0 then A(0)=0. This implies that the set of solutions contain the zero vector.
Property 2: If $x,y\in V$ then $x+y\in V$ where V is the set of solutions of the given system.
Let x and y are two solutions of the homogeneous system. Then, we have, Ax=0 and Ay=0.
Now,
$A\left(x+y\right)=Ax+Ay$=0+0=0
Hence, x+y is a solution of the homogeneous system which implies that the set of solutions is closed under addition.
Property 3: If c is any scalar and x is a solution of the homogeneous system, then cx is also a solution of the homogeneous system. Since x is a solution of the homogeneous system, Ax=0.
Now,

Hence, cx is a solution of the homogeneous system.
Therefore, the set of solutions of the homogeneous system form a vector space over $\mathbb{R}$.
The set of solutions will not form a vector space if we take the non-homogeneous system i.e., Ax=b where b contains atleast one nonzero element.
Reason: If we take x=0,
$A\left(0\right)=b$
$b=0$
but b is nonzero which implies that $A\left(0\right)\ne 0$. Therefore, the set of solutions of the non-homogeneous system does not contain the zero vector and hence, it is not a vector space.

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