This is a finite group of 8 elements and the multiplication table should give all possible products. So for example to show each element has inverses, simply list the inverse for each elelment. You can refer to your multiplication table to verify this.
You don't really need to show...
You need to show that the binary structure fits the definition of a group.
ie: show
1) multiplication is associative
2) existence of identity
3) existence of inverses
Im sorry, I can't make sense out of a single thing you said. Rather than go through your post and ask what each statement means, perhaps you could rewrite it, including definitions for any new terms you use (such as Xm). Also look up the definitions of any terms you encounter to make sure you...
Let X = [0,1) with the normal metric on R
Let Y be the uinit square, ie: Y = \{(x_1,x_2): x_1^2 + x_2^2 = 1\}, with the normal metric on R^2
Let f:X \rightarrow Y, \ \ \ \theta \mapsto (cos2 \pi \theta, sin2 \pi \theta) .
Take U = [0, 1/2). U is open in X but f(U) is not open in Y.
Oh I see, you used the product rule (integration by parts). That'll work too!
Sure, you can always substitute the dummy variable so long as its different from the one used for the limit of integration, so in this case you'd have to first evaluate the expression,
\left[...
No. One of the transformations is left multiplication by the matrix A, the other is left multiplication by the matrix B, which I defined to be (A^T)A.
I'm going to abandon this approach for the moment and start again. (I think I may be confusing you by calling A and B linear...
Your first two answers look good.
For the third, I can't make sense of what you've done. What is u'?
Heres a hint:
Define the functions F and G as
F(x) = \int_{0}^{x} \left( \int_{0}^{u} f(t)dt \right) du
G(x) = \int_{0}^{x}f(u)(x-u)du
Find the derivatives of F and G with respect...
Are you asking what the shape of a baseball field is? If so, you can find a diagram at this link:
http://en.wikipedia.org/wiki/Baseball_diamond
PS: I miss those names that you find littered throughout high school Calc and physics texts. Milt Famey :rofl:
In regards to the dimension theorem V is not to be taken as A. V is the domain of the arbitrary linear transformation T. V is a vector space. A is a matrix.
Let B = (A^t)A. So B is an mxm matrix.
In my earlier post, I was hinting that you treat both A and B as
linear transformations...
You need to show some work first. You say you've seen a proof already but are having trouble understanding it, well what is the proof and which part is giving you trouble. There is more than one way to do it.
It's a good idea to attempt the proof yourself first without looking at the...
What have you tried so far?
Have you looked for counter examples or attempted a proof?
Do you have an idea of how to prove/disprove statements like these?
it should be
\sum_{i = 1}^{k + 2} i2^i = \left(\sum_{i = 1}^{k+1} i2^i \right) + (k+2)2^{k+2}
As a simple example consider:
\sum_{i=1}^{4}i = 1 + 2 + 3 + 4 = (1 + 2 + 3) + 4 = \left(\sum_{i=1}^{3}i \right) + 4
more generally, if f is a function defined on the integers, and...
The notation "sup Un" makes no sense because Un is a real number not a set of real numbers, and sup only applies to sets.
You seem confused about what the limsup of a sequence is.
If (s_n) is a bounded sequence of real numbers, then we define
\limsup s_n = \lim_{k \rightarrow...