>>It's a group of vectors that can be multiplied by any scalar and/or >>added together in any way, > >Not just any way -- there are precise conditions that the sum must >satisfy. Look for the axiomatic definition in your book(s). It's not >at all difficult to memorize. Do it.
I already have memorized it. The space has to be closed under addition and scalar mult, and contain the zero vector. Therefore, it satisfies those 8 properties:
*v* + *w* = *x* (in the space) c*v* = c*v* (in the space) 0*v* = 0 1*v* = *v*
>The algebra is the important thing, not the picture. And your current >understanding of the algebra is insufficient to keep you from getting >confused on the upcoming test.
But, I did so well in algebra in 7th-9th grades. How much different can this be? I've got the definitions well enough: I got a B- on the first exam. It's the *concepts* since image/kernel/basis that have confused me, like I said.
>>I guess I could also really use some help with understanding how a >>mapping gets converted into a matrix, and then how to solve it. > >I like the term "linear transformation" and I think you should use it >too;
OK. See? I never knew what that term meant. Now I do.
>Do you know >what restrictions I'm talking about? f(a+b) = f(a)+f(b) and >f(ca) = cf(a) of course. That, by definition, is what makes the >mapping *linear*.
Right. And also the 0 vector.
>Try working it out for the 90-degree rotation I mentioned >above, and then try your matrix out on some 2D column vectors to see >if they really do turn 90 degrees when you multiply by the matrix.
If something is rotated 90 degrees, the first point, cosine, goes from 1 to 0. And sine goes from 0 to 1. I still don't really get how to represent that in a matrix.
Theta(pi/2) of (0,1) becomes (1,0). Does that mean the matrix is just:
/ 1| 0| \ / 2X1
>(I am assuming you multiply with matrix on the left and column vector >on the right, as is done in Schaum's outline; hopefully that is how >your prof does it too,