Strang §3.1 · Spaces of vectors
A subspace is a flat sheet through the origin.
A subspace is a set of vectors you cannot escape by adding and scaling. In the plane there are only three kinds: the origin alone, a line through the origin, or the whole plane.
v and w in S ⇒ cv + dw in S, for every c and d
Think: a train on rails
Straight rails run through the station, which is the origin. Drive forward, reverse, combine trips: you always stay on the rails. That line is a subspace.
Now picture rails that miss the station. Scale any position by 0 and you should land at the origin, but the origin is not on those rails. So they are not a subspace.
The three rules
- It contains the zero vector.
- Adding two vectors in the set stays in the set.
- Scaling a vector in the set stays in the set.
Examples and non-examples
- Yes: the line y = 2x. Add or scale points on it and you stay on it.
- No: the line y = 2x + 1. It misses the origin.
- No: the first quadrant. Multiply by −1 and you leave it.
- No: the two axes together. (1, 0) + (0, 1) = (1, 1) is on neither axis.
Worked example in 3D
All (x, y, z) with x + y + z = 0. Add two such vectors and the sums add: 0 + 0 = 0. Scale one and the sum scales: c·0 = 0. So it is a subspace, a plane through the origin.
With x + y + z = 1 instead, the origin fails the test. Not a subspace.
Why we care
The column space from lesson 1 is a subspace. So is the nullspace on the next slide. Chapter 3 is about these spaces, because they tell you exactly which equations Ax = b can be solved, and how many solutions they have.