Vector Spaces & Subspaces
You have been reading the word “space” since lesson one. Time to pin it down. A vector space is any set where adding and scaling behave the way you already expect. A subspace is a space living inside a bigger one — and it is the single definition the rest of this course leans on hardest.
What is a Vector Space?
A vector space is a set V with two operations: add two elements of V, and scale an element by a real number. Both results must land back inside V. That is the whole idea. Everything else is bookkeeping.
The Axioms
There are eight, and you already obey every one without thinking about it. Grouped, they say only this: addition and scaling do not surprise you.
- Closure — u + v and c u stay in V
- Order-free addition — commutative and associative
- A zero — 0 is in V, and v + 0 = v
- Negatives — every v has a −v
- Scaling plays fair — it distributes, and 1 v = v
Polynomials are Vectors Too
Take every polynomial of degree 2 or less: a + bx + cx². Add two of them and you get another one. Scale one and you get another one. Every axiom holds, so this set is a vector space — with no arrows anywhere in sight.
What is a Subspace?
A subspace is a subset of a vector space that is a vector space in its own right, under the same addition and the same scaling. Not any old subset — it has to be closed, so you can never operate your way out of it.
The Three-Part Test
You never re-verify eight axioms. The parent space already supplies them. You check three things, in this order.
- Contains the zero vector — is 0 in S?
- Closed under addition — u, v in S ⇒ u + v in S
- Closed under scaling — u in S ⇒ c u in S, for every real c
A Plane Through the Origin
Two vectors in ℝ³. Drag the sliders to swing v around and watch the sheet they sweep out. Flatten v onto u and the plane collapses to a line.
Worked Examples
Example 1. In ℝ³, let S be every vector whose last coordinate is zero — the xy-plane. Run the test.
- Zero? (0, 0, 0) has last coordinate 0. Yes.
- Sums? 0 + 0 = 0, so the sum stays flat.
- Scaling? c · 0 = 0, still flat.
The Non-Example
In ℝ², take the line y = x + 1. It is every bit as straight as a subspace, and just as easy to draw. It is not a subspace — and the very first part of the test kills it.
- Zero? No: 0 ≠ 0 + 1. Test over.
- Sums? (0,1) + (1,2) = (1,3), and 3 ≠ 1 + 1. It leaves.
- Rule of thumb — a line or plane must pass through the origin, or it is not a subspace.
Span
You met span as a picture in Lesson 3: the reachable space. Here is the sentence behind the picture — and the useful part is that a span is always a subspace, with no test required.
Basis & Dimension
A basis is a spanning set with nothing to spare: drop any one vector and the span shrinks. Every basis of the same space turns out to have the same number of vectors, and that number is the dimension.
Why You Will Need This
Module 3 asks which vectors a matrix crushes to zero. That set — the null space — is a subspace, and its dimension is the nullity. The matrix columns span a second subspace, whose dimension is the rank. Rank and nullity are subspace arithmetic.
Three Ways to Say How Big
A norm assigns a length to a vector. Module 11 leans on L1 and L2 constantly; the third one is just as useful and hardly ever introduced. L2 is the straight-line length from Lesson 3. L1 adds up absolute values. L∞ reports the single largest one and ignores the rest.
The Unit Ball
The unit ball is every vector of length at most 1. Slide p and watch the shape change: a diamond at p = 1, a circle at p = 2, and a square as p runs off to infinity.
What you learned
A vector space is a set closed under addition and scaling — arrows optional. A subspace is a space inside a space, and three checks settle it: zero, sums, scalings. Span is everything reachable, a basis is a spanning set with nothing spare, and dimension counts it. L1, L2 and L∞ measure size three different ways.