I am fully aware of the great volume of pages on stack exchange that ask this question; but none of the pages truly address my concern.
My concern being: Could anyone please suggest linear algebra book(s), preferably just the one, that discusses the topics delineated in the following link (I have even copied and pasted the topics of our syllabus below the primary body of this query):
https://www.isibang.ac.in/~adean/infsys/database/Bmath/LAlg2.html
The aforesaid is the link to a webpage of a college where I study in.
I am a first year undergraduate pursuing pure mathematics. And I find it to be the most painful chore to read several different books simultaneously. The mathematical narrative is lost when I do so.
I fully understand that this is how UNDERGRADUATE mathematics works. Nevertheless, to begin with, I would be more than indebted to anyone who can share ONE (or as few as possible) book(s) that would do the help me understand the topics mentioned in the aforesaid link.
Please help.
I have copied and pasted our course outline here:
Determinant of n-th order and its elementary properties, expansion by a row or column, statement of Laplace expansion, determinant of a product, statement of Cauchy- Binet theorem, inverse through classical adjoint, Cramer's rule, determinant of a partitioned matrix, Idempotent matrices. Norm and inner product on Rn and Cn, norm induced by an inner product, Orthonormal basis, Gram-Schmidt orthogonalization starting from any finite set of vectors, orthogonal complement, orthogonal projection into a subspace, orthogonal projector into the column space of A, orthogonal and unitary matrices. Characteristic roots, relation between characteristic polynomials of AB and BA when AB is square, Cayley-Hamilton theorem, idea of minimal polynomial, eigenvectors, algebraic and geometric multiplicities, characterization of diagonalizable matrices, spectral representation of Hermitian and real symmetric matrices, singular value decomposition. Quadratic form, category of a quadratic form, use in classification of conics, Lagranges reduction to diagonal form, rank and signature, Sylvesters law, determinant criteria for n.n.d. and p.d. quadratic forms, Hadamards inequality, extrema of a p. d. quadratic form, simultaneous diagonalization of two quadratic forms one of which is p.d., simultaneous orthogonal diagonalization of commuting real symmetric matrices, square-root method.
Note: Geometric meaning of various concepts like subspace and flat, linear independence, projection, determinant (as volume), inner product, norm, orthogonality, orthogonal projection, and eigenvector should be discussed. Only finite-dimensional vector spaces to be covered.
