Advanced Calculus fi..

Advanced Calculus, Fifth Edition8. Prove the rule for equality in (1.108) as stated in the theorem. [Hint: Use (1.109) toreduce the problem to that for lu. vl = lul Ivl. Use the first part of the theorem and themeaning of linear dependence to prove the desired result.]9. Prove: The vectors of an orthogonal system are linearly independent.10. Prove the following rules for vectors in Vn, with the aid of the hints given.a) Rule (a) [Hint: If clvl + ... + ckvk = 0 and, for example, ck # 0, then express vk asa linear combination of vl, ..., vk-1. For the converse, suppose vk = clvl + ... +ck-lvk-1 and write this equation in the form clvl + ... + ckvk = 0.1b) Rule (b) [Hint: Suppose vk = 0 and choose cl ,..., ck not all 0 so that clvl + ... +CkVk = 0.1C) Rule (c) [Hint: One has clvl + ... + ck+lvk+l = 0 with not all c's 0. Show thatck+~ cannot be 0.1d) Rule (d)e) Rule (e)f) Rule (f)g) Rule (h) [Hint: Apply Rules (g) and (e).]h) Rule (j) [Hint: By Rule (i), vl, ..., vk do not form a basis. Hence one can choosevk+l so that vl, ..., vk+l are linearly independent. If k + 1 < n, repeat the process.]i) Rule (k) [Hint: Apply Rule (j) to obtain a ba~is vl ,.. ., v,. By Rule (g), ul ,..., uk+l,vk+l, .... V , are linearly dependent, so thatclul + ' '' + Ck+lUk+l + ck+2Vk+l + ' ' ' + Cn+lVn = 0.Show from the hypotheses that ck+2 = 0, .... Cn+l = 0.1j) Rule (1) [Hint: Let vl = ul ,..., vk = uk. Then choose vk+l, ..., v, as in Rule (j) sothat vl, ..., v, form a basis. Now apply the Gram-Schmidt process to vl, .. ., v,.]11. a) Let vl , v2, vg be linearly independent. Prove that the vectors (1.11 1) are nonzero andform an orthogonal system. Prove also that, for h = 1,2,3, uh is a linear combinationof vl, . . . , vh, and vh is a linear combination of ul, . . . , uh.b) Cany out the process (1.1 11) in v3 with vl = i, vz = i + j + k, v3 = 2j + k, andgraph.12. Prove that if u;, ..., uz and v;, ..., v,* are the components of u and v with respect tothe orthonormal basis ef, ..., ei, then (1.116) holds.13. (Change of basis) Let e;, ..., e,T be an orthonormal basis of Vn, so that (1.114) and(1.115) hold for an arbitrary vector v. In particular, we can writeaij=ei.er, i=1, ..., n. j=l ,..., n.a) Show that, with A = (aij),col(v;, . . . , u,*) = A col(ul, . . . , u,).Thus the matrix A provides the link between the components of v with respect to thetwo orthonormal bases. [Hint: Write v = ulel +. . . + v,e, = v;e; + . . . + vie; anddot both sides with ey for i = 1, . . . , n.]b) Show that the jth column of A gives the components of ej with respect to the basisef, . .., ez and the ith row of A gives the components of e: with respect to the basisel, ..., en.c) Show that A is an orthogonal matrix (Section 1.13).