By Edwards, Charles Henry

Modern conceptual therapy of multivariable calculus, emphasizing the interaction of geometry and research through linear algebra and the approximation of nonlinear mappings through linear ones. even as, considerable realization is paid to the classical functions and computational equipment. 1000s of examples, difficulties and figures. 1973 edition.

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Extra info for Advanced calculus of several variables

Example text

V1–V8 are all immediate consequences of our definitions and the properties of . For example, to prove V6, let x = (x1, . . , xn). Then The remaining verifications are left as exercises for the student. A vector space is a set V together with two mappings V × V → V and × V → V, called vector addition and scalar multiplication respectively, such that V1–V8 above hold for all and (V3 asserts that there exists such that x + 0 = x for all , and V4 that, given , there exists such that x + (− x) = 0).

Vn for V. Since it is clear (by linearity of L) that the vectors L( v1), . . , L( vn) generate Im L, it suffices to prove that they are linearly independent. Suppose Then so t1 v1 + · · · + tn vn = 0 because Ker L = 0. But then t1 = · · · = tn = 0 because the vectors v1, . . , vn are linearly independent. 3). 2 Let L : n → m be defined by L( x) = A x, where A = (aij) is an m × n matrix. Then (a) Ker L is the orthogonal complement of that subspace of n that is generated by the row vectors A1, .

3). 2 Let L : n → m be defined by L( x) = A x, where A = (aij) is an m × n matrix. Then (a) Ker L is the orthogonal complement of that subspace of n that is generated by the row vectors A1, . . , Am of A, and (b) Im L is the subspace of m that is generated by the column vectors A1, . . , An of A. PROOF (a) follows immediately from the fact that L is described by the scalar equations so that the ith coordinate Li( x) is zero if and only if x is orthogonal to the row vector Ai. (b) follows immediately from the fact that Im L is generated by the images L( e1), .