Note 7.6.1. Unit highlights.
Big question. What remains true when vectors are matrices or polynomials, and how does an inner product determine projection and best approximation?
Notation. \(V\text{,}\) \(U\subseteq V\text{,}\) \(\mathcal P_{\le d}\text{,}\) \(M_{m,n}\text{,}\) \(\langle \mathbf u,\mathbf v\rangle\text{,}\) \(\|\mathbf v\|\text{,}\) \(d(\mathbf u,\mathbf v)\text{,}\) \(G\mathbf c=\mathbf b\text{,}\) \(A^T\mathbf r=\mathbf0\text{,}\) \(P_t\text{,}\) \(T_m\text{,}\) \(H_f(\mathbf a)\text{,}\) \(\frac12\mathbf h^T H_f(\mathbf a)\mathbf h\text{.}\)
Learning outcomes. By the end of this unit, students should be able to use U7-LO1 through U7-LO7 to work with abstract vector spaces, inner product spaces, matrix and polynomial examples, projection equations, Taylor and least-squares approximations, Gram-Schmidt in polynomial spaces, and Hessians as quadratic forms.
Concepts. Vector space, subspace, span, independence, basis, dimension, coordinate vector, inner product, norm, distance, orthogonality, Gram matrix, projection, polynomial space, Taylor polynomial, sampled least squares, Gram-Schmidt, orthogonal polynomial basis, quadratic form, Hessian.
Core results. Matrix spaces and bounded-degree polynomial spaces are vector spaces. An inner product gives norms, distances, orthogonality, and projection. Orthogonal projection depends on the inner product. Projection equations in a finite-dimensional inner product space have Gram-matrix form \(G\mathbf c=\mathbf b\text{.}\) Taylor approximation is local, while continuous and sampled least squares depend on an interval or sample points. Shrinking-interval least-squares projections recover Taylor approximation in the limit. Gram-Schmidt creates coordinates adapted to an inner product. The Hessian is the matrix of the quadratic part of a local polynomial model.
Toolbox skills. U7-LO1 through U7-LO7 use vector-space axiom checks, subspace tests, bases and coordinates, inner-product tests, norms, distances, Gram matrices, projection equations, residual conditions, Gram-Schmidt computations, and quadratic forms.
Main application. Abstract vector and inner product spaces, with polynomial spaces as the final synthesis of projection, regression, Taylor approximation, Gram-Schmidt, and Hessians.
Connections. Unit 2 supplies subspaces, bases, dimensions, coordinates, and linear maps. Unit 3 supplies local linearization and tangent planes. Unit 4 supplies projection, residual orthogonality, least squares, and Gram-Schmidt. Unit 5 supplies Hessians and quadratic forms.
Study anchors. U7-LO1 and U7-LO2: Abstract vector spaces. U7-LO3 and U7-LO4: Inner product spaces. U7-LO2 through U7-LO7: Polynomial spaces and course connections. U7-LO3 through U7-LO7: Applications and computation recap. U7-LO7: Hessians as quadratic forms.
Applications and computations readiness checklist. Given a short Unit 7 calculation, code snippet, table, or diagram, I can identify the vector space; decide whether a set is a subspace; write coordinate vectors for matrices or polynomials; compute an inner-product norm or distance; form or interpret a Gram matrix; read \(G\mathbf c=\mathbf b\) as projection equations; read a polynomial design matrix; explain what \(A^T\mathbf r=\mathbf0\) checks; compare Taylor, continuous least-squares, sampled least-squares, and shrinking-interval approximations; explain why Gram-Schmidt changes coordinates; identify sampled best-fit plane equations; and identify the Hessian as the quadratic part of a Taylor polynomial.
Common mistakes. Thinking vectors must be columns in \(\mathbb R^n\text{;}\) forgetting that the zero vector in a polynomial space is the zero polynomial; treating an affine set as a subspace; assuming “best approximation” has meaning before choosing an inner product; confusing local Taylor matching with global or sampled least squares; thinking \(A^T\mathbf r=\mathbf0\) means \(\mathbf r=\mathbf0\text{;}\) and writing displayed vector computations as row vectors when column-vector notation is intended.
