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practice_second_test_323

practice_second_test_323 - rank AB ≤ rank B Q8 A 2 × 2...

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MATH 323, PRACTICE TEST 2 Q1. Find the transition matrix from the basis B = { (1 , 0) T , (1 , 1) T } to the basis C = { (1 , 2) T , (1 , 3) T } . Q2. Find bases for the row and column spaces of A = 1 1 2 0 2 1 2 3 1 4 1 1 2 0 2 . Q3. Let L : P 3 R 2 be defined by L ( p ( x )) = p (0) p (1) . Find bases for the kernel and range of L . Q4. Find the matrix of L in Q3 when P 3 has basis { 1 , x, x 2 } and R 2 has basis { (1 , 1) T , (1 , 2) T } . Q5. Let A = 3 3 2 8 . Find the eigenvalues and eigenvectors. Q6. For the matrix A of Q5, solve the system Y 0 = AY, Y (0) = 1 - 5 . You may use any calculations done in Q5 if you wish. 1
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Q7. If A and B are n × n matrices, prove that N ( B ) N ( AB ), and that rank( AB ) rank( B
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Unformatted text preview: rank( AB ) ≤ rank( B ). Q8. A 2 × 2 matrix A has eigenvalues λ 1 = 1 and λ 2 = 2 with respective eigenvectors (1 , 1) T and (1 , 2) T . Write the vector v = (1 , 0) T as a linear com-bination of these eigenvectors and then find A 4 v . Q9. Let L 1 : V → V be a linear transformation. Define L : V → V by L ( v ) = L 1 ( L 1 ( v )) , v ∈ V. Prove that L is a linear transformation. Q10. If A and B are similar matrices, prove that A T and B T are similar. 2...
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