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Unformatted text preview: 18.06 Problem Set 1 Due Wednesday, 13 February 2008 at 4 pm in 2-106. Problem 1: Do problem 28 from section 1.1 (pg. 10) in the book. Solution (10 points) We are assuming that abcd 6 = 0, so in particular none of a,b,c,d can be 0. Now suppose that ( a,b ) is a multiple of ( c,d ). That means that for some number k we have a = kc and b = kd . Substituting in we find ( a,c ) = ( kc,c ) and ( b,d ) = ( kd,d ). Of course these are multiples: we get ( a,c ) by multiplying ( b,d ) by c d . Note that we must know that d is not 0 in order to be able to define this ratio. Problem 2: Do problem 31 from section 1.2 (pg. 20). Solution (10 points) Yes, three vectors u , v , w in the plane can have negative dot products with each other. We just need to pick three vectors so that every angle between them is more than 90 degrees. For example, take u = 1 v =- 1 2 w =- 1- 1 Problem 3: For the system A x = b (where A is a 3-by-3 matrix), choose A and b so that: 1. (row picture) the three planes meet in a common line 2. (row picture) all three planes are parallel but distinct 3. (row picture) the intersection of the first two planes does not intersect the third plane 4. (column picture) b is not a linear combination of the columns of A 1 5. (column picture) b is a multiple of the second column of A Solution (2+2+2+2+2 points) 1) There are several ways to do this. The easiest is to note that the third equation must be a linear combination of the first two (for example the sum of the first two). One example is x = 0 y = 0 x + y = 0 All three planes contain the z-axis. Thinking in terms of elimination, what we must do is find a matrix A whose upper triangular form U only has 2 pivots. Of course, this doesn’t tell us whether the planes all intersect along a line or whether they don’t intersect at all. We need an appropriate choice of b to put us in the first situation....
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- Spring '08
- Triangular matrix, Row