Question

If anyone can help me on this difficult linear algebra problem I would appreciate it!

http://dl.dropbox.com/u/6717478/Untitled.png

Answer 1

I've figured out the solution to part be
but I'm having lots of trouble with part a

Answer 2

Find a formula first for the area of a parallelogram which is described by the two vectors. Write it out and try and figure out how you would do that. I'm thinking you'll want the length of one of them multiplied by the vector component orthogonal to the first, of the second vector. There's a formula for that. Once you have that, see what happens when you throw the vectors through the linear operator. I've forgotten exactly what will happen... but I guess you should be able to use the formula for the two original vectors, just modified a bit, somehow...

Answer 3

When dealing with the vectors that are the result of applying the linear operator, remember that it's linear. So you can pull out constants, and separate bits that are just added together...

Answer 4

The area of the parallelogram is length v multiplied by length u*. u can be composed by the addition of two vectors, one that is a multiple of v, and one that is orthogonal to v. You take the part that is orthogonal to v as being u*.