Spaces within Spaces

Linear Algebra (Matrices) · Axiom Academy

When is a slice of a vector space in its own right? Three tests — and they all hinge on staying inside under addition and scaling. A nonempty subset is a subspace when it survives three checks. The third and second are the heart of it: an operation done inside W must give a result that is still inside W — that property is called closure . Watch the simplest example: a line W through the origin, . Add two of its vectors, scale one of them — the results never leave the line. closed under addition and scaling 2. Up a Dimension: A Plane Through the Origin The same test passes for a plane through the origin, . Now there are two independent directions, but closure still holds: any combination of vectors lying in the plane lands somewhere else in the plane . The sum is the diagonal of the parallelogram they span — and that diagonal is itself a combination , so it lies in the plane. Scaling tilts a vector longer or shorter along the same flat sheet. The origin is the case s=t=0 . 3. The Telltale Failure: Off the Origin Slide that line so it no longer passes through the origin — say the line y=1 in the plane, . It looks just as straight, but it is not a subspace. The same addition we did in Step 1 now leaves the set. the sum is not on the line — closure fails is present; and stay on the line. Subspace. ; adding two points of L leaves L . Not a subspace.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.