I am presented with the question:
Let $\mathbb{C}$ be the set of complex numbers. Define addition on $\mathbb{C}$ by:
$(a+bi)+(c+di)=(a+c)+(b+d)i$. Define scalar multiplication by: $\alpha\cdot
(a+bi)=\alpha \cdot a + \alpha \cdot bi$ for all real $\alpha$. Show that $\mathbb{C}$ is a
vector space with these operations.
I'm a little unclear on vector spaces which makes proving this question fairly difficult. I'm not even sure where to start. If someone could explain how to go about answering this question, that would be awesome.
Best Answer
You need to go read the axioms for a vector space (which you undoubtedly took notes on) and verify them one at a time for this example.
You have been given a description of an addition operation and a description of a scalar multiplication. At present, you have no idea if they follow the rules for vector spaces. One by one, show that the addition and scalar multiplication described above fit the description these axioms require.
We could describe the computations in excruciating detail, but you will probably learn the most if you struggle this way for a while. If you are truly stuck in 30 minutes time, then come back for a hint.
The axioms should include (among other things) axioms like:
"There is an operation $+$ such that for all $x,y\in V$, $x+y\in V$ and furthermore each $x$ has an inverse with $+$, and there is a neutral element for $+$..."
"Scalar multiplication distributes over $+$, and ..."