Linear Algebra 6 | Linear Subspaces [dark version]

Linear subspaces are the subsets of a vector space where vector arithmetic never “escapes” the set: scaling and adding vectors from the subset always stays inside it. That closure under linear combinations is what makes a subspace behave like a smaller vector space, letting calculations be done entirely within the subset rather than relying on the surrounding space. The practical payoff is clear: once a set is a subspace, it inherits the same algebraic rules as the ambient space, so linear algebra can be carried out locally on that subset.

The discussion starts with the geometric intuition in  (the plane). Lines through the origin are the canonical example of a linear subspace. If a vector lies on such a line, scaling it by any scalar keeps it on the same line, and adding two vectors on the line produces another vector on the line. By contrast, a line that does not pass through the origin fails the subspace requirement: scaling a point on that line typically moves it off the line, so the set cannot support the same vector operations indefinitely.

A formal definition is then given for a subset U of  (written as  in the transcript as RN). U is called a (linear) subspace when it is non-empty and closed under the two basic operations used to build linear combinations: scaling vectors and adding vectors. In other words, if vectors u1, u2, … are taken from U and scalars 1, 2, … are chosen, the resulting linear combination 1233 (as described via the general sum of scaled vectors) must still land back in U. This “can’t leave U by scaling or adding” property is the defining feature.

To make the concept usable in concrete problems, the transcript provides a simple three-condition test for subspaces in RN. First, the zero vector must belong to U. Second, U must be closed under scalar multiplication: for every u in U and every scalar , the product  must also be in U. Third, U must be closed under addition: for any u and v in U, the sum u+v must also be in U. If any one of these fails—especially if 0 is missing—U cannot be a subspace.

Finally, the transcript highlights “trivial” subspaces. The zero subspace (containing only the zero vector) satisfies all three conditions immediately. The entire space RN is also a subspace because scaling and addition are already guaranteed to stay within RN. These two extremes bracket all other subspaces: every non-trivial subspace lies somewhere between the smallest one (just {0}) and the largest one (RN). The next step, promised for a later part, is to work through non-trivial examples by applying the three-condition test.