課程名稱︰工程數學-線性代數 課程性質︰電機系大一必修(統一教學) 課程教師︰蘇柏青、馮世邁、林茂昭 (三個班) 開課學院：電機資訊學院 開課系所︰電機工程學系 考試日期（年月日）︰2015/04/22 考試時限（分鐘）：110分鐘 試題 : 1.(10%) Let A be an nxn matrix. Let R be the reduced row echelon form of A. Prove or disprove that det(A) = det(R). 2.(10%) Let T: R^n->R^m be a linear transformation defined by T(x)=Ax for all x 屬於 R^n, where A is an mxn matrix. Let U:R^m->R^n be a linear transformation defined by U(x)=A^(T)x for all x 屬於 R^m, where A^(T) is the transpose of A. Prove or disprove that the dimension of the range of T is the same as the dimension of the range of U. 3.(a)(6%) Let Q be an nxn invertible matrix. Let {u1,u2,...,uk} be a linearly independent set of vectors in R^n. Prove that {Qu1,Qu2,...,Quk} is linearly independent. (b)(4%) Suppose that {Pu1,Pu2,...,Puk} is linearly independent, where P is an nxn matrix . Is it necessary that k≦n? 4. Let A = [a1 a2 a3 a4 a5] be a 4x5 matrix and b 屬於 R^4. The general solution to Ax=b is given by [x1] [-5] [-2] [ 1] [x2] [ 0] [ 1] [ 0] [x3] = [-3] + x2[ 0] + x5[ 0] [x4] [ 2] [ 0] [-1] [x5] [ 0] [ 0] [ 1] (a)(3+2%) Find the rank and nullity of A. (b)(3%) Find a basis for Null A. (c)(7%) Let A'=[a2 a1 a3 a4 -a5] and b'=b+a3. Find the general solution to A'x=b' in vector form. 5. Consider the following linear transformations: [ x1+ x2] T1: R^2->R^3 defined by T1([x1]) = [ x1-3x2] [x2] [ 4x1 ] [x1] T2: R^3->R^2 defined by T2([x2]) = [ x1- x2+4x3] [x3] [ x1+3x2 ] (a)(8%) Find the standard matrices of T1, T2, T1T2, T2T1. (b)(4%) Is T1T2 onto? Is T1T2 one-to-one? (c)(4%) Let U1:R^n->R^m and U2:R^m->R^p be linear. Determine if the following statements are true or false (explain your answer): (i) If m>n, then U2U1 cannot be onto. (ii) If m<p, then U2U1 cannot be onto. (d)(4%) Let U1 and U2 be the linear transformations defined in (c). Prove that if U1 and U2 are one-to-one, then U2U1 is one-to-one. 6. [1 1 1 ] Consider the 4x3 matrix A = [1 a b ] where a,b 屬於 R(實數). [1 a^2 b^2] [1 a^3 b^3] (a)(8%) Show that the column vectors of A form a linearly set if and only if a≠1,b≠1,and a≠b. (b)(6%) Assume a＝b≠1. Find rank A and nullity A. (c)(6%) Assume a≠1,b≠1,and a≠b. Let 向量b=[0 1 a+b a^2+ab+b^2]^(T) 屬於 R^4. Show that 向量b=Col A by explicitly specifying a vector x 屬於 R^3 such that Ax=向量b. (Hint: Recall that a^3-b^3=(a-b)(a^2+ab+b^2) a^2-b^2=(a-b)(a+b) ) 7. Let A be an mxn matrix and R be its reduced row echelon form. We know that there is an mxm invertible P such that PA=R. (a)(8%) Use the formula PA=R to prove that the rows of R are linearly independent if and only if the rows of A are linearly independent. (b)(7%) If rank A=m, show that P is uniquely determined by proving that P = [ap1 ap2 ... apm]^(-1) where ap1, ap2,...,apm are the pivot columns of A. --

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