Calculus, Volume II: Multi-Variable Calculus and Linear by Tom M. Apostol

By Tom M. Apostol

Quantity II of "Calculus", contained during this paintings, provides multi-variable calculus and linear algebra, with purposes to differential equations and likelihood. quantity I, bought individually, offers one-variable calculus with an advent to linear algebra.

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That is, T maps a point with polar coordinates (r, 0) onto the point with polar coordinates (r, 0 + v), where q~ is fixed. Also, T maps 0 onto itself. 12. T maps each point onto its reflection with respect to a fixed line through the origin. 13. T maps every point onto the point (1, 1). 14. T maps each point with polar coordinates (r, 0) onto the point with polar coordinates (2r, 0). Also, T maps 0 onto itself. 15. T maps each point with polar coordinates (r, 0) onto the point with polar coordinates (r, 20).

Repeating the argument with x1 replaced by xi, we find that each cj = 0. This proves that S is independent. 7(b) shows that S is a basis for V. EXAMPLE. In the real linear space C(O,27r) with the inner product (f, g) = JiBf(x)g(x) dx, let S be the set of trigonometric functions {u,, ul, u2, . } given by %&4 = 1, uznpl(x) = cos nx, uZn(x) = sin nx, If m # n, we have the orthogonality relations 2n s0 u~(x)u,(x) dx = 0, f o r n = 1,2,.... Orthogonality in a Euclidean space 19 so S is an orthogonal set.

The null space consists of all elements in V orthogonal to z. E X A M P L E 6. Projection on a subspace S. 15). ,‘mce T(x) = s, we have T(x) = 0 if and only if x = sL . Therefore, the null space is z l, the orthogonal complement of S. EXAMPLE 7. DifSerentiation operator. The null space consists of all functions that are constant on the given interval. EXAMPLE 8. Integration operator. The null space contains only the zero function. 3 Nullity and rank Again in this section T denotes a linear transformation of a linear space V into a linear space W.

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