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

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Unformatted text preview: 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- 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 6 = 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- 8- 1 2- 8- 1- 4- 4 1 1 3 1 1 3 0 1 1 8- 1 4 0 0 0 1 3 1 0 5 8 0 1 1 8- 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 zerosand this property will be preserved under row operations. Alternatively,column is all zerosand this property will be preserved under row operations....
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.

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

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