Chapter synthesis: independence controls redundancy, span controls reach

The chapter pairs two audits: redundancy (dependence) and coverage (span). Basis vectors pass both audits simultaneously, which is why matrix columns later encode everything about a linear map .

A set that spans but is not independent can still be trimmed to a basis by removing redundancies. A set that is independent but not spanning extends to a basis by adding vectors outside its span. Dimension of a subspace equals the size of any basis for that subspace.

Professional habit: when stuck, ask separately "Do I have redundancy?" and "Do I reach the target subspace?" Span answers reach; independence answers redundancy. For a pair of arrows in $\mathbb{R}^3$, span is the plane or line you can reach; independence means neither arrow lies in the span of the other unless it is zero.

This vocabulary feeds directly into column space, null space, and rank in later chapters. The same two questions reappear whenever you read a matrix as a list of column destinations .

Check your understanding. The tasks below rest on these ideas: Correct: drop dependent generators one at a time until what remains is independent and still spans. Not quite: it does span (so it is a generating set), is nonempty, and is not yet a basis because of the redundancy. Correct: keep adding independent vectors outside the current span until it spans, completing a basis. Not quite: not spanning means it is not yet a basis, it spans more than $\{\mathbf{0}\}$, and a basis never contains $\mathbf{0}$. Correct: dimension is the (common) size of every basis of the subspace. Not quite: it is independent of drawings, is usually smaller than $n$ for a proper subspace, and has nothing to do with length. Correct: span is the reachable set, while independence is the redundancy check. Not quite: the second option swaps the two ideas, and they are distinct concepts unrelated to angles or lengths alone.