Prove isomorphism

May 2009
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Yes, I'm in need of a thorough review but I don't have the luxury of time as my test is in an hour and my final is in a week. A simple example similar to the problem I listed would suffice :p
Here is a basic outline of what you should be doing:

To prove two vector spaces \(\displaystyle U\) and \(\displaystyle V\) are isomorphic you must

(1) Find a linear map between them, \(\displaystyle T: U \rightarrow V\),

(2) Prove that \(\displaystyle T\) is an injection,

(3) Prove that \(\displaystyle T\) is a surjection.

In some ways, step (1) is the hardest - it requires some imagination. For the question you are looking at, you have done this;

\(\displaystyle T:\left( \begin{array}{cc}
a & b\\
c & d\end{array} \right) \mapsto ax^3+bx^2+cx+d\)

So this is your linear map. You must, however, prove that it is linear. That is, you have to prove that

(1.a) \(\displaystyle T(u_1) + T(u_2) = T(u_1+u_2)\),

(1.b) \(\displaystyle \lambda T(u) = T(\lambda u)\).

For (2), \(\displaystyle T\) is an injection if and only if \(\displaystyle \left(T(u_1) = T(u_2) \Rightarrow u_1=u_2 \right)\), by definition. Now, as your map is a linear map, \(\displaystyle T(u_1) = T(u_2) \Leftrightarrow T(u_1) - T(u_2) = 0 \Leftrightarrow T(u_1-u_2) = 0\). Therefore, \(\displaystyle T\) is an injection if and only if the set \(\displaystyle K=\{u: T(u) = 0\}\) consists only of the zero of the linear space \(\displaystyle U\) (as then \(\displaystyle T(u_1) = T(u_2) \Rightarrow T(u_1-u_2) = 0 \Rightarrow u_1-u_2=0 \Rightarrow u_1=u_2)\)). This set \(\displaystyle K\) is called the kernel of your linear map, \(\displaystyle T\).

Thus, what you need to prove is that if \(\displaystyle T(u) = 0 = 0x^3+0x^2+0x+0\) then \(\displaystyle u=0 = \left( \begin{array}{cc}
0& 0\\
0& 0\end{array} \right)\).

Does that make sense?

Now, for (3) it is sufficient to prove that each of the generators of the space \(\displaystyle V\) is mapped onto by some element of \(\displaystyle U\) (why?). So, what are the generators of \(\displaystyle V\) in this case, and which elements of \(\displaystyle U\) are mapped to them?

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I would agree with Tonio, you do need proper help. We can, and are willing, to help you here. However, you need to give more constructive replies. `I'm confused' doesn't really count as constructive. You have to say why you are confused.
 
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