Linear algebra done rightThis text for a second course in linear algebra is aimed at math majors and graduate students. The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presentswithout having defined determinantsa clean proof that every linear operator on a finitedimensional complex vector space (or an odddimensional real vector space) has an eigenvalue. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus, the text starts by discussing vector spaces, linear independence, span, basis, and dimension. Students are introduced to innerproduct spaces in the first half of the book and shortly thereafter to the finitedimensional spectral theorem. This second edition includes a new section on orthogonal projections and minimization problems. The sections on selfadjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text. 
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Review: Linear Algebra Done Right
User Review  GoodreadsMy goto recommendation for linear algebra. Clear, concise, and helpful. An actual math book; not another "vectors are thingies with magnitude and direction herp derp" text for electrical engineers. Read full review
Review: Linear Algebra Done Right
User Review  Jamie Banks  GoodreadsWonderful!!!! No determinants til the end. Emphasis on concept over formal, so rare in Lin alg. Weak on determinants  Artin or Halmos, Finite Dimensional Vector Spaces, are better. The only reason I understand GrahamSchmidt conceptually and not as an algorithm to be memorized!!! Read full review
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a e F additive identity additive inverse adjoint block uppertriangular matrix chapter characteristic polynomial completing the proof complex innerproduct space complex numbers complex vector space compute consisting of eigenvectors coordinates corollary corresponding decomposition definition denote dimension dimensional dimV direct sum eigenpair eigenvalues elements of F entries equation example exists finite finitedimensional vector space function hence implies injective invariant subspaces invertible isometry length linear algebra linear combination linear map linearly independent tuple matrix with respect minimal polynomial ntuple nilpotent nonzero vector Note null space orthogonal orthonormal basis orthonormal tuple permutation positive integer positive operator Prove real innerproduct space real numbers real vector space right side scalar multiplication selfadjoint operator space over F span(vi spanning tuple spectral theorem square matrix square root subset Suppose surjective tuple of vectors unique verify vT*T words z e F