How to prove that complex numbers $(\mathbb{C})$ are linear space over real numbers $(\mathbb{R})$ field?

How to prove that complex numbers $\mathbb{C}$ are linear space over real numbers field $\mathbb{R}$ ?

Solution 1:

Verify that it satisfies the axioms, of course!

1. The sum of two complex numbers is a complex number.
2. Multiplying a complex number by a real number yields a complex number.
3. Addition of complex numbers is associative.
4. Addition of complex numbers is commutative.
5. The complex number $0$ is such that $z+0 = z$ for all complex numbers $z$.
6. Given a complex number $z$, there is a complex number (namely, $-z$) such that $z+(-z) = 0$.
7. Multiplication by reals numbers is associative: $\alpha(\beta z) = (\alpha\beta)z$ for all $\alpha,\beta\in\mathbb{R}$, $z\in \mathbb{C}$.
8. Multiplication by reals distributes over the sum of complex numbers: $\alpha(z+w) = \alpha z + \alpha w$ for all $\alpha\in\mathbb{R}$, $z,w\in\mathbb{C}$.
9. Multiplication by a complex number distributes over the sum of real numbers: $(\alpha+\beta)z = \alpha z + \beta z$ for all $\alpha,\beta\in\mathbb{R}$, $z\in\mathbb{C}$.
10. Multiplication by the real number $1$ is the identity mapping: $1z = z$ for all $z\in\mathbb{C}$.

If all 10 are true, then $\mathbb{C}$ is a vector space/linear space over $\mathbb{R}$.

(More generally, if $\mathbf{F}$ is a field, and $\mathbf{K}$ is a field that contains $\mathbf{F}$, then $\mathbf{K}$ is a vector space over $\mathbf{F}$).

Solution 2:

The term "linear space" is another word for vector space. There are two aspects to showing that something (call it $V$) is a vector space over a field $K$ (in this case, $K=\mathbb{R}$, the real numbers):

First, you need to specify a vector addition operation "$\star$", that takes any two elements of $V$ and produces another one, and a scalar multiplication operation "$\circ$", that takes an element of $K$ and an element of $V$ and produces an element of $V$. Second, you need to show that those operations satisfy the axioms for being a vector space.

Here, the set $V$ under consideration is $\mathbb{C}$, the complex numbers. The vector addition is that is implied is just the usual addition of complex numbers; that is, we define $$a\star b:=a+b\text{ for all }a,b\in\mathbb{C}.$$ The scalar multiplication (i.e., multiplying a vector (an element of $V$) by a scalar (an element of $K$)) that is implied is just the multiplication of a complex number by a real number; that is, we define $$\lambda\circ a:=\lambda a\text{ for all }\lambda\in\mathbb{R}\text{ and }a\in\mathbb{C}.$$

The axioms for being a vector space are here. There is not really anything substantial you need to show. For example, $1\circ a=a$ for any $a\in\mathbb{C}$ because, by definition, $$1\circ a=1a=a$$ (because $1\in\mathbb{R}\subset\mathbb{C}$ is the multiplicative identity of $\mathbb{C}$'s usual multiplication). Another example: $$\lambda\circ (a\star b)=\lambda(a+b)=\lambda a +\lambda b=(\lambda\circ a)\star(\lambda\circ b)$$ because multiplication of complex numbers distributes over addition.