# Nonhomogeneous Systems of m equations in n unknowns and Solution Spaces.

My book says that solutions sets of nonhomogeneous systems of m equations in n unknowns is NEVER a subspace of R^n. Why? If we look at any two planes intersecting in R3, there may be a line formed. This line DOES have the 0 vector (a single point on the line), is closed under addition and multiplication.

No, the line of intersection does not contain the origin. If the system $A{\bf x}={\bf b}$ is nonhomogeneous, this means by definition that ${\bf b}\ne0$. Therefore ${\bf x}=0$ is not a solution.
• A line is a set of points, not a set of vectors. A point can be "interpreted" as a vector, in which case it is the vector from the origin to the point. So saying that the vector $\bf0$ lies on the line is the same as saying the point $\bf0$ is on the line. See if this post is helpful. – David Aug 13 '14 at 0:28