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For a compact introduction, see Artin [1]. 38 In linear algebra a basis is commonly introduced as a linearly independent set of vectors that spans the space. We mean essentially the same thing, except that for a finite dimensional vector space, we have put it differently in terms of the notion of uniqueness. This appears to be a more parsimonious way of introducing the concept at this stage. Uniqueness of expansion is equivalent to linear independence. 11) h(a)Da , H= a∈I where the coefficients h(a) are the values of its impulse response h = Hδ0 .

1, 0, 0, . . + 0, 1, 1, 1, . . = 1 + σω. Then, rearranging and using the rules for fractions, σ= 1 1−ω 28 Strictly speaking, we should use different symbols to designate equality and addition here, because they are not the same as those for sequences. But that would make the notation cumbersome. So we use the same symbols in both the cases. 29 As in the case of the convolution sign, we shall omit the product sign and write p · q simply as pq for fractions p and q. 30 Such a structure is called in algebra a field.

Algebraic Preliminaries 46 Given (nonempty) sets X and Y , X × Y , the cartesian product of the two sets, is the set: X × Y = {(x, y)|x ∈ X and y ∈ Y }. We talk of cartesian products of more than two sets in a similar fashion. For a cartesian product of the same set, say X, taken n times, we use the abbreviation X n: Xn = X × · · · × X . n times 2 3 Thus, we write R for R × R, and R for R × R × R. By a relation, unless otherwise mentioned, we mean a binary relation. Formally, a relation R from a set X to a set Y is a subset of X × Y : R ⊆ X × Y .

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An Introduction to Differentiable Manifolds and Riemannian Geom. by W. Boothby

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