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wksht10 - MATH 152 L1 L2 Applied Linear Algebra and...

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MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 10 Worksheet (April 11, 2005) : Euclidean Vector Q1. (Linear combination of vectors) Consider the vectors v 1 = (1 , 3 , 0), v 2 = ( - 1 , 1 , 1), v 3 = (3 , 1 , - 1). Is b = (3 , 5 , - 1) a linear combination of v 1 , v 2 , v 3 ? Q2. (General solution in vector form) Consider A = 2 0 2 1 1 1 - 1 0 0 1 - 2 1 , b = 2 - 1 - 2 . (a) Find a special solution x = ( x 1 , x 2 , x 3 , x 4 ) of Ax = b satisfying x 1 = x 2 = 0. (b) Solve the homogeneous equation Ax = 0 . (c) Solve the equation Ax = b . Q3. (Free variables and rank) Suppose Ax = b is a consistent system of 5 equations in 9 variables. If the rank of A is 3, how many free variables are there in the general solution? Q4. (Null space) Find the null space of A if A = - 3 6 - 1 1 - 7 1 - 2 2 3 - 1 2 - 4 5 8 - 4 .
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Q5. (Span) Consider the vectors v 1 = (1 , 2 , 3 , 4), v 2 = (2 , 3 , 4 , 1), v 3 = (3 , 4 , 1 , 2). (a) Is b = (0 , 0 , 1 , - 1) in the span of v 1 , v 2 , v 3 ? (b) Do v 1 , v 2 , v 3 span R 4 ? If not, how many more vectors do you need to add in order to span R 4 ? Explain. Q6. (Span) Consider the vectors v 1 = (0 , 1 , 2 , - 1), v 2 = (1 , 4 , - 3 , 1), v 3 = (2 , 3 , 1 , 1). v 4 = ( - 1 , 3 , 0 , 2). Determine whether the vector b = (1
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