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Let $V$ be a finite dimensional vector space over a field $\Bbbk$. I want to show that $End(V)$ is isomorphic to the ring $M_{n}(\Bbbk)$ of $n \times n$ matrices with entries in $\Bbbk$. So I need to show that the ring homomorphism $\phi:End(V) \rightarrow M_{n}(\Bbbk)$ is an isomorphism; i.e, there is a ring homomorphism $\theta: M_{n}(\Bbbk) \rightarrow End(V)$ such that $\theta(\phi(\alpha))=\alpha$ for all $\alpha \in End(V)$ and $\phi(\theta(A))=A$ for all $A \in M_{n}(\Bbbk)$.

Can I assume $\phi:End(V) \rightarrow M_{n}(\Bbbk)$ exists or do I need to prove this first and how do I go about proving that it is an isomorphism?

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    First of all, you'd need to assume something about $V$ ... or specify how $n$ is related to $V$ – 2017-02-27
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    @HagenvonEitzen Sorry I forgot to add that! – 2017-02-27
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    The answer to "Can I assume... or do I need to prove this first" always depends on context, but if you don't see this fact, I definitely recommend working it out for yourself. As for how to prove this: Given a linear transformation $\phi: V \to V$, what choice do we need to make to produce a matrix representation $[\phi]$ of $\phi$? (Presumably the problem is referring to the ring isomorphisms produced by the standard way of doing this.) – 2017-02-27

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Well, you should start with the definition of $\phi$. Constructions are generally preferred over existance. Especially in this case I don't see a way to prove this without explicite definition.

So let $V$ be a $n$-dimensional vector space over a field $k$. Pick a basis $\{e_1,\ldots,e_n\}\subset V$. Now define two functions:

$$i_k:k\to V$$ $$i_k(\lambda)=\lambda e_k$$

$$\pi_k: V\to k$$ $$\pi_k(\lambda_1 e_1+\cdots +\lambda_n e_n)= \lambda_k$$

So the first one is the embedding onto $k$-th coordinate and the second one is the projection onto $k$-th coordinate. They are both linear.

Now define

$$\phi:\mbox{End}(V)\to\mathbb{M}_{n\times n}(k)$$ $$\phi(f)_{ij}=(\pi_i\circ f\circ i_j)(1)$$

It is easy to see that $\phi$ is linear. A bit more calculations have to be done to show that $\phi$ preserves ring multiplications (with composition on $\mbox{End}(V)$ and matrix multiplication on $\mathbb{M}_{n\times n}(k)$). And all that is left is to show that either it is "1-1" and "onto" or explicitely define the inverse map. Can you complete the proof?

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No, you can't assume that $\phi\colon\operatorname{End}(V)\to M_n(\Bbb k)$ exists. In particular, unless you explicitly state what $\phi$ is, you will hardly be able to show that there is an iverse $\theta$; after all, there do exists non-isomorphisms between these two rings.

Start by making use of $\dim V=n$. What does that mean explicitly? How can you use that to describe an endomorphism with a matrix??