Question

Determine whether the given set S is a subspace of the vector space V.

A. V=Rₙ, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix.

Answer 1

Set S, defined as the solutions to a homogeneous linear system Ax=0, is a subspace of V=Rₙ because it includes the null vector, is closed under vector addition, and closed under scalar multiplication, meeting all the subspace criteria.

Step-by-step explanation:

The student has asked to determine whether a set S, defined as the set of solutions to a homogeneous linear system Ax=0 where A is a fixed m×n matrix, is a subspace of the vector space V=Rₙ. To verify if S is a subspace, we must check if it is closed under vector addition and scalar multiplication, and if it includes the null vector. Since S contains solutions to Ax=0, it includes the null vector which is the solution when x is the zero vector. Moreover, for any two vectors x and y in S, A(x+y)=Ax+Ay=0+0=0, thus x+y is also in S. Similarly, for any scalar c and any vector x in S, A(cx)=c(Ax)=c⋅0=0, hence cx is in S. Therefore, S fulfills all the conditions to be a subspace of V=Rₙ.