2700rev - A = 7 0 4 3 0 3-5 0 1 8 Consider the system of...

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MATH 2700 - Exam 1 - Sample Problems 1. True or False: If a system of linear equations has two different solutions, then it must have infinitely many solutions. Explain your answer. 2. Solve the following system of equations: x 1 - x 2 + 2 x 3 = 3 3 x 1 - 4 x 2 + 5 x 3 = 9 x 1 + x 2 + x 3 = 6 3. Let v 1 ,..., v r R n be vectors. Suppose that u and w can be written as linear combinations of v 1 ,..., v r . Prove that u + w is also linear combination of the v 1 ,..., v r . 4. Let A = 0 1 1 2 1 5 1 1 3 and b = 5 1 3 . Solve A x = b and express your answer in vector parametric form. 5. Let u = 1 - 1 - 2 , v = 3 1 - 6 , and w = - 2 2 6 . Decide if w is in span { u , v } . 6. Are u = 0 0 1 , v = 1 1 0 , and w = 1 1 1 linearly independent? 7. Without using any row operations decide if the the columns of A are linearly independent or dependent, where
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Unformatted text preview: A = 7 0 4 3 0 3-5 0 1 . 8. Consider the system of equations: 2 x 1-4 x 2-2 x 3 = b 1-5 x 1 + x 2 + x 3 = b 2 7 x 1-5 x 2-3 x 3 = b 3 Describe the set of vectors b for which this system does have a solution. 9. Is the map T : R 3 → R 3 defined by T ( x 1 ,x 2 ,x 3 ) = ( x 2 x 3 ,x 1 x 3 ,x 1 x 2 ) a linear transformation? Why or why not? 1 10. Let T : R 3 → R 3 be the linear transformation given by T ( x 1 ,x 2 ,x 3 ) = ( x 1 ,-x 2 ,x 3 ) find the standard matrix of T. 11. ( Challenge ) If T is a one to one linear transformation and u , and v are linearly independent, show that T ( u ), and T ( v ) are linearly independent. 2...
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This note was uploaded on 06/14/2011 for the course MATH 2700 taught by Professor Atim during the Spring '08 term at North Texas.

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2700rev - A = 7 0 4 3 0 3-5 0 1 8 Consider the system of...

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