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Linear Algebra
Step 6 of 22

Subspaces and dimension

Understand first: linear independence

Spans keep producing the same shapes: a line through the origin, a plane through the origin, or everything. Never a curve, never a shape that misses the origin. That pattern has a reason and a name.

What qualifies

A subspace is a set closed under the only two operations that exist here: add two vectors from it, and you stay inside; scale one, and you stay inside.

Test a line through the origin, say all multiples of (1,2)(1,2). Add 3(1,2)3(1,2) and 5(1,2)5(1,2) to get 8(1,2)8(1,2) — still on the line. Scale any of them — still on the line. It qualifies.

Now test a line that misses the origin, the points where y=x+1y = x + 1. Take (0,1)(0,1) and (1,2)(1,2), both on it. Their sum is (1,3)(1,3), and 3≠1+13 \neq 1 + 1. It fell off. Not a subspace.

Every subspace must contain the origin, because scaling any of its vectors by zero lands there. That single requirement is why "through the origin" keeps appearing.

Dimension is a count you cannot fake

The dimension of a subspace is the number of vectors in a basis for it: how many independent directions it holds.

  • A line through the origin: dimension 1.
  • A plane through the origin: dimension 2.
  • All of 3D space: dimension 3.
  • The origin alone: dimension 0, a subspace with nothing to move along.

The remarkable part, and the reason dimension is trustworthy: every basis for a given subspace has the same number of vectors. You may describe a plane with {(1,0,0),(0,1,0)}{(1,0,0),(0,1,0)} or with {(1,1,0),(1,−1,0)}{(1,1,0),(1,-1,0)}, but never with three independent vectors, and two are never enough for 3D space. The count is a property of the space itself.

Act I is complete

You now have the objects: vectors, the two operations, span, independence, basis, subspace, dimension.

Everything so far has been static — describing what is. Act II sets it in motion, and asks a question that turns out to be the whole subject: what happens if you take all of space and move it at once?