Linear Algebra: A Geometric ApproachCRC Press, 1993. gada 15. maijs - 384 lappuses This is an undergraduate textbook suitable for linear algebra courses. This is the only textbook that develops the linear algebra hand-in-hand with the geometry of linear (or affine) spaces in such a way that the understanding of each reinforces the other. The text is divided into two parts: Part I is on linear algebra and affine geometry, finishing with a chapter on transformation groups; Part II is on quadratic forms and their geometry (Euclidean geometry), including a chapter on finite subgroups of 0 (2). Each of the 23 chapters concludes with a generous helping of exercises, and a selection of these have solutions at the end of the book. The chapters also contain many examples, both numerical worked examples (mostly in 2 and 3 dimensions), as well as examples which take some of the ideas further. Many of the chapters contain "complements" which develop more special topics, and which can be omitted on a first reading. The structure of the book is designed to allow as much flexibility as possible in designing a course, either by omitting whole chapters or by omitting the "complements" or specific examples. |
Saturs
Vectors and vector spaces | 3 |
Matrices | 13 |
Systems of linear equations | 25 |
Some linear algebra | 43 |
Rank | 65 |
Determinants | 73 |
Affine space I | 93 |
Affine space II | 106 |
Transformation groups | 196 |
Euclidean Geometry | 213 |
Bilinear and quadratic forms | 215 |
Diagonalizing quadratic forms | 231 |
Scalar product | 238 |
Vector product | 254 |
Euclidean spaces | 259 |
Unitary operators and isometries | 278 |
Geometry in affine planes | 118 |
Geometry in 3dimensional affine space | 130 |
Linear maps | 145 |
Linear maps and matrices affine changes of coordinates | 163 |
Linear operators | 179 |
Isometries of planes and threedimensional space | 295 |
Diagonalizing symmetric operators | 310 |
A Domains rings and fields | 329 |
Bibliography | 357 |
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a₁ affine plane affine space affine subspace affine transformation associated vector subspace automorphism axioms b₁ basis e1 bijection called canonical basis Cartesian equations coefficients column convex angle coordinates Corollary defined definition denoted det(A determined diagonal diagonalizable dim(S dim(V dim(W direction vector e₁ eigenvalue eigenvector element elementary End(V equivalent Euclidean space Euclidean vector space example finite dimensional follows immediately geometric vectors given GL(V GLn(K Hermitian homogeneous identity integer inverse Isom(E isometry isomorphism K-vector ker(F l₁ linear combination linear map linearly independent matrix Mn(K multiplied n-tuple non-zero orthogonal orthonormal basis parallel permutation perpendicular polynomial Proof properties Proposition Prove quadratic form rank reflexion respect rk(A rotation satisfies scalar product skew-symmetric space of dimension subgroup subset Suppose symmetric bilinear form Theorem unitary operator V₁ V₂ vector subspace W₁ X₁ zero