Notes on Linear Algebra for Polynomials

A structured walkthrough of linear algebra concepts applied to real polynomials of degree n. Covers proof that polynomials form a vector space by verifying all axioms, explains linear independence, span, and basis with the monomial basis as a concrete example, demonstrates how to check if an arbitrary set of polynomials forms a basis using row-echelon form, defines an inner product via integration over finite bounds and verifies its axioms, and introduces orthogonality for polynomials noting that the monomial basis is not orthogonal under this inner product.