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hw1 - Linear Algebra Homework 1 Selected Solutions Any...

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Linear Algebra Homework 1 Selected Solutions Any mistakes are the sole responsibility of the author-T.Peters 1.1.6g Represent as a matrix and row reduce: 1 3 2 3 2 - 1 1 2 3 2 3 2 1 2 2 12 5 1 10 1 2 6 - 3 2 4 3 3 5 20 24 1 1 2 6 - 3 0 0 - 9 9 0 10 - 6 16 1 2 6 - 3 0 0 1 - 1 0 5 - 3 8 1 2 6 - 3 0 1 0 1 0 0 1 - 1 1 2 0 3 0 1 0 1 0 0 1 - 1 1 0 0 1 0 1 0 1 0 0 1 - 1 So x 1 = - 1 , x 2 = - 1 , and x 3 = 1. 1.1.9 This whole problem can be interpreted geometrically: we have two lines l 1 and l 2 defined by the equations. The system has a unique solution if and only if the two lines are not parallel, ie m 1 = m 2 . If m 1 = m 2 then the only way for the system to be consistent is for the lines to be coincident, ie be the same line. This happens only when b 1 = b 2 . 1.2.8b Represent as a matrix and row-reduce: 1 3 1 1 3 2 - 2 1 2 8 3 1 2 - 1 - 1 1 3 1 1 3 0 - 8 - 1 0 2 0 - 8 - 1 - 4 - 4 1
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1 3 1 1 3 0 1 1 8 0 - 1 4 0 0 0 1 3 1 0 5 8 0 0 0 1 1 8 0 - 1 4 0 0 0 1 3 So that x 1 = - 5 8 t , x 2 = - 1 4 - 1 8 t , x 3 = t , and x 4 = 3. 1.2.9 (a) It is not possible for the system to be inconsistent since this only happens when then RREF of the matrix has a row of the form(0 0 0 | 1) but this is not possible since the 4th column is all zeros–and this property will be preserved under row operations. Alternatively, we could realize that this is a homogeneous system, and homogeneous systems are always consistent since they have the trivial solution.
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