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 $\mathbb{R}$ .

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Suppose you [ have a consistent system of linear equations, with coefficients in R, which are homogeneous - that is, all the

Who knows?

Haiden Frazier

Beginner2022-02-23Added 10 answers

It is given that we have a consistent system of linear equations with coefficients in

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

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

Let x and y are two solutions of the homogeneous system. Then, we have, Ax=0 and Ay=0.

Now,

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

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,

but b is nonzero which implies that