Algebra Through Practice: Volume 4, Linear Algebra: A Collection of Problems in Algebra with Solutions
Problem-solving is an art central to understanding and ability in mathematics. With this series of books, the authors have provided a selection of worked examples, problems with complete solutions and test papers designed to be used with or instead of standard textbooks on algebra. For the convenience of the reader, a key explaining how the present books may be used in conjunction with some of the major textbooks is included. Each volume is divided into sections that begin with some notes on notation and prerequisites. The majority of the material is aimed at the students of average ability but some sections contain more challenging problems. By working through the books, the student will gain a deeper understanding of the fundamental concepts involved, and practice in the formulation, and so solution, of other problems. Books later in the series cover material at a more advanced level than the earlier titles, although each is, within its own limits, self-contained.
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Duality and normal transformations
Solutions to Chapter 1
Solutions to Chapter 2
Test paper 1
Algebra bases block Chapter characteristic polynomial choose clearly commute complex Consequently Consider Conversely corresponding Deduce define determine diagonal diagonalisable differentiation dimension distinct dual eigenvalue eigenvectors elements equal equation example exists f relative fact False field F Find Find a Jordan finite finite-dimensional vector space follows geometric multiplicity given gives Hence holds integer invertible matrix isomorphism Jordan basis Jordan normal form Ker f Kert linear functional linear transformation linearly independent LIV,V matrix of f minimum polynomial non-singular non-zero observe obtain orthogonal orthogonal matrix positive definite possible projection Prove quadratic form question relative represented respect result satisfies Show similar similarly solution solve space of dimension spanned squares standard basis subset subspace Suppose symmetric symmetric matrix t-invariant true unique vector space whence zero