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, Gram-Schmidt, and Hessians.
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:
Core outcomes.
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U7-LO1. Verify vector-space properties and produce counterexamples when a property fails.
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U7-LO2. Work in nonstandard vector spaces, especially matrix spaces and polynomial spaces, using subspaces, span, independence, bases, dimension, and coordinates.
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U7-LO3. Verify inner-product properties, compute norms and distances, and explain how changing the inner product changes geometry.
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U7-LO4. Derive projection equations in finite-dimensional inner product spaces, including Gram-matrix equations \(G\mathbf c=\mathbf b\text{.}\)
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U7-LO5. Compare Taylor, continuous least-squares, and sampled least-squares approximations by their residual conditions and error measures.
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U7-LO6. Apply Gram-Schmidt in polynomial spaces and interpret the resulting orthogonal polynomial basis.
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U7-LO7. Interpret Hessians as quadratic forms and connect this interpretation back to optimization.
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; run Gram-Schmidt; and interpret quadratic forms.
Application and computation outcomes. Interpret Lab U7
[U7-LO3, U7-LO4, U7-LO5, U7-LO6, U7-LO7 | C+M+T | Core] by reading polynomial feature matrices, least-squares coefficients, fitted values, residual checks, shrinking-interval coefficients, Gram-Schmidt checks, sampled plane fits, and Hessian quadratic forms.
