solutions

# solutions - Midterm 1, Math 20F - Lecture B (Spring 2007)...

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Midterm 1, Math 20F - Lecture B (Spring 2007) 1. Consider the system of linear equations 2 x 1 + 4 x 2 - x 3 = 2 x 1 + 2 x 2 + hx 3 = b where h and b are scalars. a) (2.5 points) Explain why the system cannot have a unique solution. Solution: The augmented matrix associated with this system can have at most two leading entries, since it has two rows and each row can contain at most one leading entry. This means that the augmented matrix has at most two pivot columns. Therefore the system must have at least one free variable. The system can be inconsistent, if the last column of the augmented matrix is the pivot column. Otherwise, if the system is consistent, the system has a free variable implying the system has inﬁnitely many solutions. b) (2.5 points) For what values of h and b is the system inconsistent? Solution: The augmented matrix for this system is ± 2 4 - 1 2 1 2 h b ² . Adding - 1 / 2 times the ﬁrst row to the second one yields ± 2 4 - 1 2 0 0 h + 0 . 5 b - 1 ² The system is inconsistent if the last column is a pivot column. This is the case whenever h + 0 . 5 = 0 and b - 1 6 = 0 , or equivalently the system is inconsistent whenever h = - 0 . 5 and b 6 = - 1 .

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2. The solution set of the homogeneous system Ax = 0 where A is a 3 by 3 matrix (corre- sponding to a system of three equations in three unknowns) is given by span - 1 1 0 , - 1 0 1 . Let p = 2 3 - 1 be a particular solution satisfying Ax = b for some given vector b of size 3. a) (2 points) Give a solution of the system Ax = b not equal to the particular solution p . Solution: Any solution for the system Ax = b is in the form p + v h where p is the particular solution and v h is a solution for the homogeneous system Ax = 0. You can pick v h any vector in span - 1 1 0 , - 1 0 1 . In particular v h = - 1 1 0 + - 1 0 1 = - 2 1 1 is a solution for the homogenous system and p + v h = 2 3 - 1 + - 2 1 1 = 0 4 0 must be a solution for the system Ax = b .
b) (3 points) Based on the solution for the homogeneous system given above provide a reduced echelon matrix that is possibly row-equivalent to

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## This note was uploaded on 03/12/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.

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solutions - Midterm 1, Math 20F - Lecture B (Spring 2007)...

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