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Additional info for Advanced Calculus: Problems and applications to science and engineering

Example text

I) A spanning set has no less than d elements. (ii) An independent set has no more than d elements. Proof. Part (i) is of course just the definition, so we need only consider part (ii). proof amounts to a reduction to the case where V is R", and an application of The Proposition 10. ,wd span V; since V has dimension d there exists such vectors. V. Then we can write Suppose, as in (ii), that vt, , vk are independent vectors in . each Vj as a vj = . linear combination of Wi, 2 flj'w, . . , wd; 1 <, j <, k j=i for suitable numbers a/.

A , basis for the kernel of T. v, , vv+i, . . , v thus span Now the crux of the matter is this: n. v+ 1, of T. Once this is shown, we will have the range w form a basis for desired the is which equation. n v, Then there is a v e R" such that w T(v). Expand v in the Let w e R(T). , , = 54 Linear Functions 1. basis vi, . . v: v , vi=T(v) = = = = c'vi + \- c"\ T(c1Yi + c'^vi) + + c"\) --- + cT(v) + c"w justified since T is linear T(vv+i) wv+i, The second line is = and the third follows since Vi, vv are Thus these last vectors w.

10. Give an example of a subset of a Cartesian product which is not a rectangle. 11 . :3* + 7y 4 (8, \),L:x-y -\ = 12. Find the line (a) (b) (c) P = P = V = 14 = to L- = = (0,-l),L:y-2x = 3 PROBLEMS 9. We can the vector X - define the line through P and Q P is parallel to the vector P parallel if and only if one is the set of all X such that Show that two vectors are of the other. Conclude that the line - a multiple Q. 2 through P and Q is the Numbers, Notation, and Geometry 27 set {P + t(P~Q):teR} 10.