# Prove isomorphism

#### Swlabr

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 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.

• tonio and Defunkt