課程名稱︰工程數學-線性代數 課程性質︰系定必修 課程教師︰馮蟻剛 馮世邁 蘇柏青 鄭振牟 林茂昭 開課學院：電機學院 開課系所︰電機系 考試日期（年月日）︰2011/04/20 考試時限（分鐘）：100 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : USE OF ALLAUTOMATIC COMPUTING MACHINES IS PROHIBITED 1.If the 4x4 matrix M satisfies the following equation, then what is the value of the det(M^3)?　　　　　　　　　　 　　　　　　　　　　　　　　　　　 (10%) ┌ ┐ ┌ ┐ │1 -2 -1 2│ │4 1 -2 -1│ │0 2 1 3│ │0 8 0 0│ │0 0 -1 1│M =│7 0 2 3│ │0 0 0 4│ │8 -7 1 4│ └ ┘ └ ┘ 2.Let F and G be two similar nxn matrices. Find two bases X and Y for R^n such that [Tf] = F and [Tf] = G (10%) X Y 3.Given two subspaces V and U of R^n, let Z = V交集U, the intersection of V and U. (a) Prove that Z is a subspace of R^n with dimension no greater than those of V and U. (b) When n=4, V = Null(A) and U = Null(B), where A and B are listed below, find a basis for Z. (10%) HINT: Z = { Z屬於R^4 | Az=0 and Bz=0 } (c) When n=4, V = Col(C) and U = Col(D), where C and D are listed below,find a basis for Z. (10%) HINT: Z = { Z屬於R^4 | z=Cx=Dy for some x屬於R^3 and y屬於R^2}. ┌ ┐ ┌ ┐ ┌ ┐ ┌ ┐ A = │-1 0 3 1│ B = │1 1 0 1│ C = │ 0 3 6│ D = │ 7 -4│ │ 1 2 -1 5│ │1 3 2 7│ │-1 -1 -3│ │-2 1│ └ ┘ └ ┘ │ 1 -3 -5│ │-4 1│ │ 3 2 7│ │ 1 1│ └ ┘ └ ┘ 4.(30%)Consider the two functions f and g defined as follows. ┌ x ┐ f:R^3→R^2, f(│ y │) = (┌ x+y+1 ┐) └ z ┘ └ z ┘ ┌ s ┐ g:R^2→R^3, g(┌ s ┐) = (│ s-1 │) └ t ┘ └ s+t ┘ (a) Find function composition h1 = g(f(.)) and h2 = f(g(.)).Your answers should include specifing their domains and codomains. ( 4%) (b) Determine whether each of f,g,h1 and h2 is an onto function and whether it is a one-to-one function (True or false only: no explanation needed.) ( 8%) (c) Determine whether each f,g,h1 and g2 is a linear transformation. Explain your answers. (12%) (d) For any of f,g,h1 and h2 that is a linear transformationm find its standerd matrix, and determine whether the matrix is invertible. 5.(20%)Conside the matrix A =┌1 1 1 ┐屬於R^4x3 │1 a b │ │1 a^2 b^2│ └1 a^3 b^3┘ (a) Show that the column vectors of A form a linearly independent set if and only if a不等於1,b不等於1 and a不等於b. (10%) (b) Assume a=b不等於1. Find rank A and nullity A. (5%) (c) Assume a不等於1,b不等於1,and a不等於b, Let ak be the kth column of A and B ={a1,a2,a3} is an ordered basis for Col A.Let v=[0 1 a+b a^2+ab+b^2]^T 屬於R^4.Show that V 屬於 Col A, and find [v]B, the coordinate of v with respect to the subspace Col A. 打到一半斷線的悲劇... -- 沒有伴隨著痛苦的教訓是沒意義的。 人如果不犧牲一些東西，就無法得到任何東西。 但是當超越了障礙，並且把得到的東西變成屬於自己的東西時... 人應該就能夠得到無法取代的鋼之心靈吧。 <Fullmetal Alchemist> --

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