Math 110.201: Linear algebra (Spring 2008)STUDY SHEET FOR THE FINAL EXAMScope of the examChapters 1–8 (minus§7.6), plus the additional topics covered in lecture. The final exam is a comprehensiveexam covering the content of the entire course.Some additional practice problemsIn addition to the problems from the midterm review sheets, try the following problems to get additionalhands-on practice; any difficulties that you may have with them will help pinpoint gaps in your understand-ing/knowledge. An answer key is not provided because understanding how to check the correctness of yoursolution is in itself a very good gauge of your grasp of the material. If you are really stuck, ask a friend, orvisit your TA or the instructor during office hours.Be sure that you know how to do all previous homework and test problems that you either couldn’t do,or hadn’t done correctly. I also suggest going through some of the true/false problems at the chapter ends.1.IfAis a 3×4 matrix such thatA===,then what must the numbersa,b,cbe? (Are the numbers even constrained by these equations?) How largecould the dimension of the kernel ofAbe?
2.Find a 3×3 matrixAwhose 2-eigenspace is spanned by [1,1,1]>, [1,0,1]>and whose 3-eigenspace isspanned by [2,1,0]>.3.Compute the determinant of the following matrices:A=0001002003001234,0-AI-I,A-AI-I.4.Let~x= [-1,1]>,~y= [2,1]>. If possible, find a 2×2 matrixAthat satisfies the following equalities:~x·A~x= 6,~x·A~y= 0,~y·A~x=-6,~y·A~y= 3.(Something to think about: Can you compute~e1·A~e2, for example? How is this number related toA?)5.Find the least square solution to the inconsistent system of equationsx= 1,x= 2,