Tips for HW #6
#4 asks you to decide whether a set of vectors spans 3-dimensional Euclidean space (R^3) and, if not, to come up with a vector not in the span. For the former, keep in mind that you know how to check whether a set of vectors is linearly independent. You also know about the deletion lemma (for chucking vectors from linearly dependent sets) and the fact that the number of vectors in any basis for a vector space is (always) the dimension of the vector space. By the way, when it comes to deleting vectors, make sure that you delete only vectors having a nonzero coefficient for some specific linear dependence. Those that have a coefficient of 0 are important and can not be discarded. As an example, consider (1,0,0), (0,1,1), (0,1,2), (0,2,3). The first vector doesn't depend on the others at all, so you cannot remove it (by the deletion lemma)!
For the other part of #4 (finding a vector outside the span), it may be instructive to write down a general linear combination of the vectors in the set (e.g., av+bw for variables a, b and vectors v and w) and trying to construct a vector that does not fit the pattern.
#1c asks for a linear combination of some vectors. That just means that you should write the vector as a sum of vectors, each of which has a coefficient in front of it. In this particular case, the coefficients will be variables and the vectors will be filled with numbers only....
#2 is a bit different than anything we talked about in class, but hopefully it isn't so bad. To get you started, call the number of sheep "s" and the number of cows "c." Then, you should be able to write down some equations which represent the constraints mentioned in the problem. Let me know if you have trouble. By the way, this problem comes from China between 2100 and 3000 years ago!
#3 complements the part of the proof that we did in class on Wednesday, August 27. Check out the proof technique part of the book - that may come in handy. In general, take your time, and try to write out (in English!) what you are looking for, then what you know. After that, you may see a path to get from what you know to what you are trying to prove. In any case, try to write enough that somebody else can follow your reasoning - I would rather see too many words than not enough! It takes a while to learn how to write a proof, so certainly let me know if you have questions.
Also for #3, there is a very nice version both of what we did in class and your homework problem in the book, on pages 13-15. If you have any trouble understanding what the author writes, please don't hesitate to contact me. Also, please write up the proof in your own words (and using the notation that I gave you in the problem), rather than just copying what Beezer says.
For the RREF problems (#4-6), try following the Gaussian elimination algorithm from class OR one of them in the book (pages 33-34, I think). I have posted a very detailed example of using Gaussian elimination, right here.