Subspaces: lines and planes through the origin

The three rules

A set S of vectors is a subspace when three things hold:

• The zero vector is in S.
• If v and w are in S, so is v + w.
• If v is in S, so is cv for every number c.

In short: every combination cv + dw of vectors in S stays in S. Mathematicians say S is closed under adding and scaling: you can't get out by doing those two things.

Stretching stays on the line

Take the line y = x/2 through the origin. The vector v = 21 is on it, since 1 = 2/2. Stretch it by any number and it stays on the line:

Adding stays on the line

Now add two vectors that are both on the line, say 21 and −4−2:

Any mix of them lands on the same line. That line passes all three tests, so it is a subspace.

Yes and no in the plane

• Yes: the line y = 2x. 12 + 24 = 36, and 6 = 2·3 ✓. Sums and multiples of its points stay on it.
• No: the line y = 2x + 1. It misses the origin. Also 01 and 13 are on it, but their sum 14 is not, since 2·1 + 1 = 3, not 4.
• No: the first quadrant (x ≥ 0, y ≥ 0). 12 is in it, but −1·12 = −1−2 is not.
• No: the two axes together. 10 and 01 are in the set, but their sum 11 is not.

So in the plane only three things pass: the origin alone, a line through the origin, and the whole plane. Why nothing else? If a subspace holds one nonzero vector, it holds all its multiples: a whole line. If it also holds a vector pointing in a different direction, then combinations of those two already reach every point of the plane (lesson 1, slide 1). So it is the origin, one line, or everything.