By Edwards, Charles Henry

ISBN-10: 0486683362

ISBN-13: 9780486683362

**Read Online or Download Advanced calculus of several variables PDF**

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**Additional info for Advanced calculus of several variables**

**Example text**

N)). 2 SUBSPACES OF n In this section we will define the dimension of a vector space, and then show that n has precisely n − 1 types of proper subspaces (that is, subspaces other than 0 and n itself)—namely, one of each dimension 1 through n − 1. In order to define dimension, we need the concept of linear independence. The vectors v1, v2, . . , vk are said to be linearly independent provided that no one of them is a linear combination of the others; otherwise they are linearly dependent.

3 that is known as the Gram–Schmidt orthogonalization process, summarized by the equations defining the orthogonal basis w1, . . , wn in terms of the original basis v1, . . , vn. Example 7 To find an orthogonal basis for the subspace V of 4 spanned by the vectors v1 = (1, 1, 0, 0), v2 = (1, 0, 1, 0), v3 = (0, 1, 0, 1), we write Example 8 Let denote the vector space of polynomials in x, with inner product defined by By applying the Gram–Schmidt orthogonalization process to the linearly independent elements 1, x, x2, .

Vk is a vector of the form a1 v1 + · · · + ak vk, where the . The span of the vectors is the set S of all linear combinations of them, and it is said that S is generated by the vectors v1, . . , vk. Example 1 n is a subspace of itself, and is generated by the standard basis vectors since (x1, x2, . . , xn) = x1 e1 + x2 e2 + · · · + xn en. Also the subset of n consisting of the zero vector alone is a subspace, called the trivial subspace of n. Example 2 The set of all points in n with last coordinate zero, that is, the set of all , is a subspace of n which may be identified with n−1.

### Advanced calculus of several variables by Edwards, Charles Henry

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