Worksheet2

Worksheet2 - to be 1 m [ m 1 v 1 + ... + m n v n ] Suppose...

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Worksheet 2 January 26th, 2009 1. Determine if b is in the span of a 1 , a 2 , and a 3 . (a) a 1 = ± 1 0 ² , a 2 = ± 2 4 ² , a 3 = ± 1 8 ² , b = ± - 1 1 ² (b) a 1 = 1 - 1 0 , a 2 = 0 1 2 , a 3 = 5 - 6 8 , b = 2 - 1 6 2. Find the general solution of A x = b . (a) Express the solution algebraically and identify the free and basic variables. (b) Express the solution set as a sum of vectors. (c) Describe the solution set geometrically. A = 1 2 - 1 3 - 2 - 3 - 5 - 14 3 8 - 16 - 4 , b = 4 - 29 - 14 3. Construct a 3 × 3 matrix not in Echelon form whose columns do not span R 3 . Show it has the desired property. 4. Let v 1 , v 2 , ..., v n be points in R k . Suppose we have a mass at each of those three points, and the weight of each mass is m 1 , m 2 , ..., m n . Let m = m 1 + m 2 + ... + m n . The center of mass of the system is defined
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Unformatted text preview: to be 1 m [ m 1 v 1 + ... + m n v n ] Suppose there are three objects, located at (1 , 1), (2 , 1), and (7 , 1), with masses 2, 6, and 2. Where do you place a fourth object with a mass of 2 so that the center of mass will be (3 , 2)? 5. Let A be a 3 × 4 matrix, let y 1 and y 2 be vectors in R 3 , and let w = y 1 + y 2 . Suppose y 1 = A x 1 and y 2 = A x 2 for some vectors x 1 and x 2 in R 4 . What fact allows you to conclude that system A x = w is consistent? 1 6. Let b be a vector in R n . If the matrix equation A x = b is consistent for all n × k matrices A what is b ? 2...
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This note was uploaded on 04/02/2009 for the course MATH 54 taught by Professor Chorin during the Spring '08 term at Berkeley.

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Worksheet2 - to be 1 m [ m 1 v 1 + ... + m n v n ] Suppose...

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