Because h(c1 v1 + + cn vn ) = c1 h(v1 ) + + cn h(vn ) and because h(0V ) = 0W , we have that c1 h(v1 ) + + cn h(vn ) = 0W with some nonzero ci . QED When is independence not lost?

2.31 List all pairs (rank(h), nullity(h)) that are possible for linear maps from R5 to R3 . 2.32 Does the dierentiation map d/dx : Pn Pn have an inverse?

1.15 Prove that for 22 matrices, the determinant of a matrix equals the determinant of its transpose. Does that also hold for 33 matrices? 1.16 Is the determinant function linear Is det(xT +y S) = xdet(T )+y det(S)?

3 The park discussed above is partially fenced so that now, every year, only 5% of the animals from inside of the park leave (still, about 1% of the animals from the outside nd their way in). Under what conditions can the park maintain a stable population now?

Prove that any two planes with n holes are homeomorphic, i.e., the location of disks D1 , . . . , Dn does not affect the topological type of R2 n Int Di . i=1 10.26.

24.7x. Prove that C(I, I) is not homeomorphic to C (pw) (I, I).

A. Prove the following: (a) If f and g both are one-to-one, then so is h; (b) If f and g both are onto, then so is h; (c) If f and g both are invertible, then so is h, and we have h1 = f 1

One way to determine if an argument is valid or invalid is to write is symbolically, and see if it is the form of one of the arguments on the previous two slides. If the argument fits the form of a valid argument, it is valid. If the argument fits the form of an invalid argument, it is invalid....

What is the future value of this deposit. (It will be compounded eight times.) (d) Do the same for the rest of the deposits: (e) How much money will be in Amy"s account on June 1, 2009? 3. Bruce starts an ordinary annuity to save for retirement. At the end of each month, $450...

44 R: And then just put in a because at 0 it"s just going to give nought for each of these. [Pause] 45 R: Right so a5 can come out of that. [Pause] 46 S: Now what sort of position do you think your solution is in?