M 346 - Old Exams Spring 2011

M 346 - Old Exams Spring 2011 - M341 Midterm Exam 1 56215...

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M341 Midterm Exam 1 56215 February 17 Name: UTEID: Score: I II III IV V VI VII You can quote theorems, lemmas, or results in books and homework assignment without proving. 1
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I.(15 points) Examples. 1. Give an example of a 3 × 3 matrix A such that A is symmetric and trace( A ) = 0 but A is not diagonal. 2. Give an example of two square matrices, A and B , that don't commute. 3. Describe completely every matrix that is both upper triangular and skew-symmetric. 2
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II. (10 points) Mark the following statements either "True" or "False". Circle T for true and F for False. No proof or explanation is needed. 1. ( T F ) [3 , 5 , 2] and [6 , 10 , 5] are parallel. 2. ( T F ) The converse and inverse of a statement are logically equivalent. 3. ( T F ) The negation of " A and B " is "not A and not B ." 4. ( T F ) Let A and B be two matrices. If AB = O , then A = O or B = O . 5. ( T F ) Consider a system of m linear equations in n variables. If m < n , then the system always have in nitely many solutions. 3
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III.(15 points) Let x 1 , . . . , x m be a mutually orthogonal set of nonzero vectors in R n . Use induction to show that ± ± ± ± ± m i =1 x i ± ± ± ± ± 2 = m i =1 x i 2 IV.(10 points) Solve real numbers a , b , c such that [ 1 b 0 2 a ][ 1 2 2 1 ] + [ 1 c 0 0 ] T = [ 2 2 3 4 ] . 4
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V. (15 points) Suppose a system of linear equations AX = B have two distinct solutions X 1 and X 2 . Prove the following statements. 1. The system AX = B has an in nite number of solutions. 2. X 1 + X 2 is also a solution of the system AX = B if and only if the system is homogeneous. 5
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VI. (20 points) Prove or disprove the following statements: 1. For all x and y in R n , x y ∥ ≤ ∥ x + y . 2. Let X , Y and Z be three n × n matrices. If XZ = YZ and Z ̸ = O , then X = Y . 3. Let a and b be two nonzero vectors. Then proj a b ∥ ≤ ∥ b . 4. Let A and B both be n × n matrices. If ( AB ) T = BA , then A and B are both symmetric. 6
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Solve the following linear systems ( a ) x 1 + 2 x 3 x 4 = 3 x 1 2 x 3 + 2 x 4 = 5 3 x 1 2 x 2 3 x 4 = 11 + x 2 + 3 x 3 + x 4 = 3 ( b ) x 1 + 2 x 3 x 4 = 0 x 1 2 x 3 + 2 x 4 = 1 3 x 1 2 x 2 3 x 4 = 2 + x 2 + 3 x 3 + x 4 = 3 by Gauss-Jordan elimination. Write down the complete solution set for systems ( a ) and ( b ) . 7
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M 346 - Old Exams Spring 2011 - M341 Midterm Exam 1 56215...

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