Chapter 15 — Key Takeaways

The big ideas

  • A basis is independent AND spanning. It is the minimal "measuring stick" for a space: spanning makes it enough to reach every vector, independence makes it not too much (no redundant rulers). Drop either condition and you no longer have a basis. Equivalently, a basis is a maximal independent set and a minimal spanning set simultaneously.

  • Dimension is the number of vectors in any basis — and the count is well-defined. Every basis of a space has the same size (the replacement/invariance theorem), so "the dimension" is an intrinsic invariant of the space. It equals the number of independent directions, i.e. the degrees of freedom. $\dim(\mathbb{R}^n) = n$; a plane through the origin has dimension 2; $\{\mathbf{0}\}$ has dimension 0.

  • Coordinates are an address relative to a chosen basis. Once you fix an ordered basis $\mathcal{B}$, every vector has exactly one coordinate vector $[\mathbf{v}]_{\mathcal{B}}$ — existence from spanning, uniqueness from independence. The same vector has different coordinate lists in different bases; the arrow never moves, only its bookkeeping.

  • A vector is not its list of numbers. The list is what you get after measuring the vector against a basis. The standard basis is special only because, in it, the address equals the entries — which is exactly why it hides the distinction this chapter exists to expose.

  • Finding coordinates = solving a linear system. Put the basis vectors in the columns of $B$ and solve $B\mathbf{c} = \mathbf{v}$; the solution $\mathbf{c} = B^{-1}\mathbf{v}$ is the coordinate vector. For a $k$-dimensional subspace of $\mathbb{R}^m$, $B$ is $m \times k$ and the system is consistent iff $\mathbf{v}$ lies in the subspace. Always verify by reconstruction.

Skills you gained

  • Test whether a set of vectors is a basis (check independence and spanning — and use the counting shortcut: in an $n$-dimensional space, $n$ independent vectors automatically span, and $n$ spanning vectors are automatically independent).
  • Compute the coordinate vector of a vector relative to any basis by solving $B\mathbf{c} = \mathbf{v}$, by hand (back-substitution / elimination) and with np.linalg.solve.
  • Find the dimension of the span of a set of vectors via the rank of the stacked matrix (np.linalg.matrix_rank), and read off a basis from the row reduction.
  • Recognize when a vector is not in a subspace (the coordinate system is inconsistent) versus genuinely in it (redundant equations, unique coordinates).
  • Re-express the same vector against two bases and explain the resulting discrepancy.
  • Implement coordinates(v, basis) in the toolkit, reusing gaussian_elimination from Chapter 4 and verifying by reconstruction.

Terms to know

basis · dimension · coordinate vector $[\mathbf{v}]_{\mathcal{B}}$ · standard basis $\{\mathbf{e}_i\}$ · ordered basis · spanning set · linear independence · finite-dimensional · degrees of freedom · coordinate isomorphism · replacement (exchange) theorem · the dimension formula $\dim(U+W) = \dim U + \dim W - \dim(U\cap W)$ · well-defined · maximal independent set / minimal spanning set

How this connects to the recurring themes

  • Theme 1 (linear algebra is the study of transformations; the matrix is just a representation). This chapter supplies the prerequisite: you cannot say "the matrix changes when the coordinates change" until you know what coordinates are. A vector's coordinate list is basis-dependent — the seed of the entire change-of-basis story.
  • Theme 2 (geometry and algebra are one object). Dimension is at once a geometric count (independent directions / degrees of freedom) and an algebraic one (number of basis vectors = rank of a matrix). The anchor — a plane in $\mathbb{R}^3$ — shows both faces in one picture.
  • The four fundamental subspaces (Part III's organizing idea). Rank is now revealed as the dimension of the column and row spaces, and nullity as the dimension of the null space; rank–nullity (Chapter 14) is pure dimensional bookkeeping. Every subspace in Strang's diagram has a basis and a dimension, computable from one row reduction.

Forward references

  • Chapter 16 (Change of Basis) is the direct sequel: it builds the change-of-coordinates matrix (glimpsed here as $B^{-1}$), shows how the matrix of a transformation transforms under a change of basis, and brings back the 2D visualizer to re-grid a transformation — the doorway to similarity.
  • Chapter 17 (Linear Regression) and Chapter 19 (Orthogonal Projection) handle the case this chapter set aside: when a vector is not in a subspace, you project it onto the subspace and coordinatize the projection.
  • Chapters 20–21 (Gram–Schmidt, orthogonal matrices) explain why orthonormal bases make coordinates trivial (each coordinate is a dot product) and length-preserving — the easy bases we kept previewing.
  • Part V (Diagonalization) is the search for the one special basis — an eigenbasis — in which a transformation's matrix is as simple as possible (diagonal). The whole payoff of "choose the right basis" begins there.
  • Threshold concept to carry forward: a vector is not its numbers; the numbers are an address relative to a chosen basis. Internalize this and the rest of the book — similarity, diagonalization, the SVD, change of representation in quantum mechanics — becomes the art of choosing the basis that makes the bookkeeping simplest.