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.
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.
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