My book says that solutions sets of nonhomogeneous systems of $m$ equations in $n$ unknowns is NEVER a subspace of $\Bbb R^n$. Why? If we look at any two planes intersecting in $\Bbb R^3$, 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.
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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.
David
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I don't really understand the concept of a subspace anymore. I thought we were supposed to look at if THE VECTOR 0 is contained in the line, not the coordinate point. I thought a subspace is a set of vectors... I'm confused. – yolo123 Aug 12 '14 at 14:48
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The line is a line... It should contain the vector 0, a 0-dimensional vector. – yolo123 Aug 12 '14 at 15:19
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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 00:28