>>I already bought the text, the Cliff's >>Notes guide to linear algebra, a 2003-version of the Schaum's outline, >>and I even have an old 1968 version of Schaum's that my grandmother >>used when she majored in math. > >This is not your grandmother's Linear Algebra course! >(I've always wanted to say something like that.)
LOL! But Schaum sure seems to think it is. There's barely any real update between the two versions.
>Sample question >(a) Prove that the set M of all n by n matrices is a vector space (using > familiar matrix addition and scalar multiplication.)
Then show A + B (above) = B + A, which is true because (a11 + b11) = (b11 + a11), same for all (aij + bij) = (bij + aij), since addition of scalars is commutative.
Then show (A+B) + C = A + (B+C) (associative).
Then I'd multiply the zero matrix by A and show that it equals 0 matrix, which is also in M.
Multiply I^n by A, to show that I*A = A.
multiply s(A) and show that:
sa11 ... sa1n . . . san1 ... sann
is also in M.
(st)(A) = (s(tA))
And, there's the eight, right? I didn't expand everything out. But, trust me, I know how to do all the computation involved with this. This was the stuff on the first test, that I did decently on.
>What is its > dimension?
N, I guess, since it has N rows. So, in homogenous form, there is a set of N equations each with N variables, right?
>(b) Prove that the map f(x) = x^t is a linear transformation from M to M
For any matrix A in set M (same A as above), f(x) = x^t maps every element aij in A to bij = (aij)^2 in matrix B. Since any element in R squared is also in R, matrix B must also be contained in set M.
For any aij in A, and any scalar s, f(saij) = (saij)^2. Again, any scalar times any other scalar in R must also be a scalar in R. So (saij)^2 = (bij) exists in B, which is included in M.
f (A + B) = (A + B)^t. Again, each element aij and bij in the matrices is in R. So, (aij + bij)^t must be in set M too.
>What is its kernel?
>(c) Compute the eigenvalues of f and find the eigenspaces.
I don't know how this would work without actual numbers and an actual matrix. Since the example is an nXn matrix, we don't know how to calculate the determinant. We need det (A-lambda*I) to find the eigenvalues and eigenvectors.
>I am of course deliberately choosing a question which emphasizes the >proper use of terminology and abstraction, but this is a perfectly >reasonable exam question. (IMHO -- but I have a reputation for thinking >"interesting" questions are reasonable so maybe you shouldn't trust me.)
No, Dave, on the contrary, I like your questions. They're *very* similar to the types that my professor would include. This may be what I mean when I say I find Schaum's and Cliff's Notes pretty much useless for my class. The exam questions are so much more abstract and technical than most of what's found in them. Like I've said, I'm alright with the computation. But, stuff like the examples you've given really, really stump me. That's why, I'm glad you offered some examples. Between Schaum's, Cliff's, and anything else I've found on the net, this is the closest I've seen examples come to being like those that my prof gives.
I guess our text book has similar examples, and Schaum's does sometimes, but in both cases the lack of clear explanations and solutions is what bothers me....