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By T. Apostol

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K + 1 . x2, . . , xk . By the induction hypothesis, the k elements ciy, - yi must be dependent. Hence, for some choice of scalars t,, . . , tk+l, not all zero, we have kfl iz2ti(ciYl - Yi) = O 9 from which we find But this is a nontrivial linear combination of y,, . . +l which represents the zero element, so the elements y1 , . . r must be dependent. This completes the proof. 8 Bases and dimension DEFINITION. for V if S is independent and spans V. The space V is called$nite-dimensional if it has a jinite basis, or if V consists of 0 alone.

Consider the three functions Ul, u2, u, given by Y+(t) = 1 + t. u,(t) = 1 , u2(t) = t 7 Prove that two of them are orthogonal, two make an angle n/3 with each other, and two make an angle n/6 with each other. 10. In the linear space P, of all real polynomials of degree 5 n, define (f3g) = -$ f(3 g(k). k=O (a) Prove that (f,g) is an inner product for P,, . (b) Compute (f, g) whenf(t) = t and g(t) = at + b . (c) Iff(t) = t , find all linear polynomials g orthogonal tof. Il. In the linear space of all real polynomials, define (f,g) = sr e&f(t)g(t) dt .

C, such that x = clel + * . * + c,e,, so we have k x - x = 2 i=l ktr ciei - 2 ciei i=kfl = 0 . 2) are indepecdent, this implies that all the scalars ci are zero. 3) are independent. Note: If V is infinite-dimensional, then at least one of N(T) or T(V) is infinitedimensional. 4. 4 Exercises In each of Exercises 1 through 10, a transformation T: V, - + VZ is defined by the formula given for T(x, y), where (x, y) is an arbitrary point in VZ . In each case determine whether Tis linear. If T is linear, describe its null space and range, and compute its nullity and rank.

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