3.1.5

Consider the vector space $$\mathbb{R}^3 $$, and let W be a subspace. We claim that W is one of the following: the zero vector alone, a line through the origin, a plane through the origin, or the whole space.

Since W is a subspace, it must always contain the zero vector 0. If it does not contain anything else, then it is the zero vector alone, W = {0}. If it does contain a vector $$w_1 \ne 0 $$, then since W is closed under scalar multiplication, W contains all vectors of the form $$cw_1 $$, where c is a real number. So W contains a line through the origin.

If W does not contain any other vectors, then it is indeed a line through the origin. Otherwise, W contains some other vector $$w_2 $$ which is not a scalar multiple of $$w_1 $$ (implying $$ w_1$$ and $$w_2 $$ are linearly independent), and since W is closed under addition and scalar multiplication, W contains all vectors of the form $$c_1w_1 + c_2w_2 $$. That is, W contains a plane through the origin.

If W does not contain any vectors other than those in the plane, then it is a plane through the origin. Otherwise, W contains yet another vector $$w_3 $$ not in the span of $$w_1 $$ and $$w_2 $$, so that $$w_1, w_2 $$ and $$w_3 $$ form a linearly independent set of vectors in W and hence in $$\mathbb{R}^3 $$. $$ \mathbb{R}^3$$ is a 3-dimensional vector space, so $$w_1,w_2 $$ and $$w_3 $$ form a basis of $$ \mathbb{R}^3$$ and hence span it, showing that W is in fact the whole space.