If C is the set of complex numbers, how do you show that C is a vector space with given operations?

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.

• Do you have the definition of vector space in front of you? There is a list of properties that have to hold for these operations, and you are supposed to verify that they do all hold. – Jonas Meyer Sep 19 '13 at 19:44
• Somebody approved an edit to this post that left intact the uncouth use of as asterisk for ordinary multiplication in MathJax. I had clicked on "improve", but when I tried to save the edit, the proposed edit had already been approved. – Michael Hardy Sep 19 '13 at 19:50
• Well thanks to whoever fixed the format on the question. Much easier to read now. – rphello101 Sep 19 '13 at 22:56

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 ..."

• Ah the axioms are key... thanks. I'm having trouble understanding the more abstract concepts so I didn't make that connection... – rphello101 Sep 19 '13 at 22:57

You must demonstrate that $C$ is a vector space by showing that it satisfies the following axioms:

1. Associativity of addition
2. Commutativity of addition
3. Identity element of addition
4. Inverse elements of addition
5. Distributivity of scalar multiplication with respect to vector addition
6. Distributivity of scalar multiplication with respect to field addition
7. Compatibility of scalar multiplication with field multiplication
8. Identity element of scalar multiplication.

Good luck!