課程名稱︰線性代數 課程性質︰電機系必修 課程教師︰蘇柏青 開課學院：電資學院 開課系所︰電機工程學系 考試日期（年月日）︰2015/4/22 考試時限（分鐘）：100 分鐘 試題 : =============================================================================== LINEAR ALGEBRA: Midterm Examination 2015 1. (10%) Let A be an n x n 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 ∈ R^n, where A is an m x n matrix. Let U: R^m → R^n be a linear transformation defined by U(x) = A^T x for all x ∈ R^n, 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 n x n 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 n x n matrix. Is it necessary that k <= n? 4. (15%) Let A = [a1 a2 a3 a4 a5] be a 4 x 5 matrix and b ∈ 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 that 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. (20%) Consider the following linear transformation: x1 x1 + x2 T1: R^2 → R^3 defined by T1( [ x2 ] ) = [ x1 - 3x2 ] 4x1 x1 x1 - x2 + 4x3 T2: R^3 → R^2 defined by T2( [ x2 ] ) = [ x1 + 3x2 ] x3 (a)(8%) Find the standard matrices of T1,T2,T1T2 and 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 linear transformation defined in (c). Prove that if U1 and U2 are one-to-one, then U2U1 is one-to-one. 1 1 1 6. (20%) Consider the 4 x 3 matrix A = [ 1 a b ] where a,b ∈ R. 1 a^2 b^2 1 a^3 b^3 (a)(8%) Show that the column vector of A form a linearly independent set if and only if a ≠ 1, b ≠ 1, and a ≠ b. (b)(6%) Assume that a = b ≠ 1. Find the rank A and nullity A. (c)(6%) Assume a ≠ 1, b ≠ 1, and a ≠ b. Let b = [0 1 a+b a^2+ab+b^2]^T ∈ R^4. Show that b ∈ Col A by explicitly specifying a vector x ∈ R^3 such that Ax = b.(Hint: Recall that a^3 - b^3 = (a-b)(a^2+ab+b^2) and a^2 - b^2 = (a-b)(a+b).) 7. (15%) Let A be an m x n matrix and R be its reduced row echelon form. We know that there is an m x m 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 = [ a_p1 a_p2 … a_pm ]^-1, where a_p1,a_p2,…,a_pm are the pivot columns of A. ================================== 試題完 =================================== 備註： 電機系大一線性代數為統一出題。 --

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