If you are preparing for the UPSC Mathematics Optional, the exam frequently asks for direct proofs of standard theorems (like the Rank-Nullity Theorem or Cayley-Hamilton Theorem). Practice writing these out exactly as presented in the book.
: Advanced chapters address bilinear forms and their associated quadratic forms, which are essential for understanding conic sections and optimization in multiple variables. Eigenvalues and Eigenvectors linear algebra by ar vasishtha pdf
Linearly dependent and independent sets, spanning sets, bases of a vector space, and the dimension of a space. If you are preparing for the UPSC Mathematics