Verifying Subspaces

Linear Algebra (Matrices) · Axiom Academy

Testing the three subspace conditions on a line through the origin Let — the set of all points on the line y=2x . Decide whether W is a subspace of by checking the three defining conditions. A subset is a subspace exactly when all three hold: The line y=2x runs straight through the origin, so the zero vector lies on it — a promising sign for a subspace. passes all three checks, so it is a subspace of . Keep these in mind: All three are required: a subspace must contain , be closed under addition, and be closed under scalar multiplication — any one failing is enough to disqualify it. Check first: it is the fastest test. If , you can stop immediately — W is not a subspace. Closure means "stays inside": adding or scaling vectors of W must land you back in W . Here both reduce to the same line equation y=2x . Contrast — a shifted line fails: is the same line moved up by 1 . The origin gives , which is false, so and V is not a subspace. The geometric rule: a line (or plane) in is a subspace exactly when it passes through the origin. This three-condition test works for any subset of — pick a candidate set, run the checks in order, and let the algebra decide.

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