In the second edition of this popular and successful text the number of exercises has been drastically increased (to a minimum of 25 per chapter); also a new chapter on the Jordan normal form has been added. These changes do not affect the character of the book as a compact but mathematically clean introduction to linear algebra with particular emphasis on topics that are used in the theory of differential equations.
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Representing linear transformations by matrices
12bis More on representing linear transformations by matrices
Systems of linear equations
The elements of eigenvalue and eigenvector theory
Inner product spaces
The spectral theorem and quadratic forms
Jordan canonical form
some numerical examples
Matrices and linear transformations
Applications to linear differential equations
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Axiom bases basis A1 Calculate the matrix called Chapter characteristic polynomial coefficient matrix column complex numbers compute defined by T(x Definition denoted dependent det(A diagonalizable dim ker dimension dimensional eigenspace eigenvalue eigenvectors endomorphism find a basis Find the matrix finite-dimensional inner product finite-dimensional vector space formula Fun(S function given by T(x hence inner product space integer invariant subspace invertible isomorphism Jordan normal form Ker(T kernel Let A1 linear combination linear equations linear extension linear span linear subspace linear system linear transformation linear transformation given linearly independent linearly independent set matrix relative multilinear form nilpotent transformation nonsingular Note numbers a1 obtain ordered basis orthogonal orthonormal basis pair plane PROOF Proposition real numbers respect scalar multiplication scalar product set of vectors Show square matrix standard basis subspace of f Suppose symmetric system of linear T-invariant Theorem transformation whose matrix vector addition vectors A1 zero