Unit 7 Abstract vector spaces and polynomial approximation
Earlier units worked mostly with vectors in \(\mathbb R^n\text{.}\) This unit introduces vector spaces and inner product spaces as abstract settings where the same algebraic and geometric ideas still work. Matrices and polynomials are the main examples. Polynomial spaces then serve as a synthesis example: after an inner product is chosen, projection connects least squares, Taylor approximation, and Gram-Schmidt.
Big question. What remains true when vectors are matrices or polynomials, and how does an inner product determine projection and best approximation?
Learning outcomes. By the end of this unit, students should be able to:
-
U7-LO1. Verify vector-space properties and produce counterexamples when a property fails.
-
U7-LO2. Work in nonstandard vector spaces, especially matrix spaces and polynomial spaces, using subspaces, span, independence, bases, dimension, and coordinates.
-
U7-LO3. Verify inner-product properties, represent inner products using matrices, compute inner products, norms, and distances, normalize vectors, test orthogonality, and explain how changing the inner product changes geometry.
-
U7-LO4. Form Gram matrices and derive, solve, and interpret projection equations in finite-dimensional inner product spaces, including \(G\mathbf c=\mathbf b\text{.}\)
-
U7-LO5. Construct and compare Taylor, continuous least-squares, and sampled least-squares approximations using their residual conditions and error measures, including comparisons on shrinking intervals.
-
U7-LO6. Apply Gram-Schmidt in polynomial spaces and interpret the resulting orthogonal polynomial basis.
Toolbox skills. Verify vector-space axioms; test subspace properties; write bases, dimensions, and coordinate vectors in matrix and polynomial spaces; test inner-product properties; compute norms and distances from inner products; form Gram matrices; solve projection equations; compare residual conditions; and run Gram-Schmidt.
