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Unformatted text preview: (Partial) Solutions to Homework 1 In the following, as a learning tool, I try to give you some of the thought process for the answers or tips on the problems and not just the answers them- selves. I’ll try to do this for the first couple homework assignments. I will not give the answers to all the problems, but if there is one that you’d really like to know, come by my office hours. Problem 2.2 Consider the augmented matrix: 2- 1 3- 6 3 1 Give a geometric reason that the associated system of equations has no solution. Given a general augmented matrix a b x c d y , can you find a condi- tion on the numbers a,b,c and d that create the geometric condition you found? Answer: In order to give a geometric reason, I should think about the geo- metric interpretation of systems of linear equations that we talked about in class. If it’s a two by two matrix, then we’re talking about lines. Maybe I can already guess what the answer should be, but let’s turn this augmented matrix into two equations of lines to be sure... The system above is equivalent to the system 2 x- y = 3 and- 6 x + 3 y = 1. We can put these lines into slope-intercept form, and we will get y = 2 x- 3 and y = 2 x + 1 3 . Since the lines have the same slope but different intercepts, they are parallel lines and so do not have any points of intersection; therefore, the system has no solutions. Now we’re asked to show something for general augmented matrices rather than just an example. But part (a) gives a hint – a two by two system has no solutions if it represents parallel lines. How can we get parallel lines in general? One way would be to look at how we solved the equations of lines earlier to get the slope – we moved the x term to the other side, and then divided by the coefficient of the y term. What would that look like in the general case? (Note: the x and y we used in part (a) are different from the...
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- Spring '10
- Linear Algebra, Elementary matrix, row equivalence