A subspace is a Subset of R^n that satisfies the following the three properties:
Non-emptiness
The zero Vector is in V. This effectively means that you cannot have a system of equations that dictate a space that does not go through it’s own local origin. For example,
Is not a subspace, as the line never goes through the origin.
This is also true of the unit circle for example:
Is also not a subspace of R^2, as the origin is not included.
Closure under addition
If u, v are vectors in V, then u + v is also in V. You can check this by adding two variants of a Vector in a set and seeing if the result is of the same form as the original. Because subspaces and spans are closely related, generally thinking of it as if the addition of two elements is in the span of the original can also help conceptually.
Closure under multiplication
This is effectively the same as the above, and is generally the reason why subspaces of the form x < y fail to be a subspace, since multiplying by -1 inverses that relationship.
Column Space
The column space of a Matrix is the subspace spanned by the vectors read by column of A. It is written Col(A).
Null Space
The null space of A is the subspace of R^n consisting of all solutions of the homogeneous equation Ax = 0: