Algebra, zadanie nr 5
ostatnie wiadomo¶ci | regulamin | latex
| Autor | Zadanie / Rozwi±zanie |
| natashq postów: 1 |  2010-03-29 15:21:27Witam, studiuje w kopenhadze, w jezyku ang, mam ogromny problem gdyz moje dziedziny to biologia z matematyka mialam doczynienia ok 6 lat temu i to byly podstawy, niestety by ukonczyc pewien etap moich studii musze zrobic tak zwane przedmioty z grupy M czyli matematyka i fizyka - jeden i i drugi przedmiot to dla mnie czarna magia...dotychczas w miare rozumialam co sie dzieje i zrobilam kilka zadan sama ale ostatnie 3 sa dla mnie bardzo trudne Pilnie ale to pilnie potrzebuje pomocy gdyz musze je oddac tak naprawde jutro ale moge to przeciagnac do czwartku... ponizej jest tresc zadan niestety po angielsku srednio mi idzie tlumaczenie tego na polski: Exercise A. Let F : Rn -> Rp be any LINEAR function and consider the subspace U c (z trzema - ponizej c ) Rp. By F−1(U). we mean the set of all vectors that F maps in U, meaning F−1(U) = {x e Rn | F(x) e U}. a) Show F−1(U) is a subspace in Rn. [Note that F does not have to be injective (1-1).] Hint: Let x1 and x2 be any two elements in F−1(U). We need to show x1 + x2 e F−1(U). There exists y1 and y2 such that F(x1) = y1 e U and F(x2) = y2 e U. Use that the sum of the y0s are in U since U is a subspace. If you can get F(x1 +x2) e U you are done (why?). Then the first condition for F−1(U) being a subspace is fulfilled. b) Let U = span{(0, 1, 1)} and F(x) = 2 3 1 0 2 1 0 0 1 (x) Find F−1(U). c) Same question as in b), with U = span{(0, 1)} and F(x) = 2 3 1 0 2 1 (x) d) In the general case a) we consider u e F−1(U) and v e N(F) - the nullspace of F. Show that any linear combination of u and v will be an element of F−1(U). How does this correspond to the observations from b) and c)? Exercise B. The linear function G : R2 -> R2 is given by G(x1, x2) = (−x2,−x1). Geometrically it represents a reflection in the line x1 + x2 = 0. a) Find the null space and the image of G. (The image of G is the collection of vectors we get from applying G on all elements of the domain of G) b) Find G(U), where U is an arbitrary one dimensional subspace of R2. c) Which subspaces are fixed spaces, meaning G(U) = U, and what is the geometric interpretation of this? Exercise C. The linear function H : R3 -> R3 is in the standard basis given by the matrix 1 0 0 2 −1 −2 0 −4 1 We now change basis for R3 using the basis vectors b1 = (1, 0, 1), b2 = (0, 1,−2) and b3 = (0, 1, 1). Find the matrix corresponding to H in the b-basis, and give a geometric interpretation. czekam na wasze komentarze rozwiazania...mozecie tez wyslac mi maila na wampircia@wp.pl |
| zorro postów: 106 | 2010-04-10 09:40:29 |
| strony: 1 | |
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