A Course in Linear Algebra
Suitable for advanced undergraduates and graduate students, this text offers a complete introduction to the basic concepts of linear algebra. Interesting and inspiring in its approach, it imparts an understanding of the subject's logical structure as well as the ways in which linear algebra provides solutions to problems in many branches of mathematics.
The authors define general vector spaces and linear mappings at the outset and base all subsequent developments on these concepts. This approach provides a ready-made context, motivation, and geometric interpretation for each new computational technique. Proofs and abstract problem-solving are introduced from the start, offering students an immediate opportunity to practice applying what they've learned. Each chapter contains an introduction, summary, and supplementary exercises. The text concludes with a pair of helpful appendixes and solutions to selected exercises.
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apply calculation canonical basis Chapter characteristic polynomial coefﬁcients complex numbers compute consider coordinate vector Corollary deﬁned deﬁnition denoted det(A determinant diagonal entries diagonalizable differential equations dim(Ker(T dim(V dim(W dimension distinct eigenvalues eigenspace eigenvalues eigenvectors Example Exercise ﬁeld F ﬁnd ﬁnite ﬁnite-dimensional vector space ﬁrst ﬁxed free variables functions f geometric Hence Hermitian inner product Im(T induction injective integer invertible isomorphism Jordan canonical form Jordan form Ker(T linear algebra linear combination linear equations linear mapping linear transformation linearly independent mathematics minimal polynomial nilpotent nonzero obtain orthogonal projection orthonormal basis proof properties Proposition Prove real numbers result roots satisﬁes scalar multiplication Section set of solutions set of vectors Show solving Span(S spectral theorem standard basis statement subset subspace surjective symmetric matrix system of equations system of linear unique zero