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In this lecture, we got a brief introduction to complex numbers, and were given the definition of a vector space (along with some examples).

*1*Complex numbers¶

$$\mathbb C = \{a + bi \mid a, b \in \mathbb R, i^2 = -1 \}$$

Note that $\mathbb R \subset \mathbb C$.

We can then define the operations of addition and multiplication in the standard way.

Exercise: Prove that $\mathbb C$ is a field. (Need to show commutativity, associtivity, existence of identities, and existence of inverses for both operations. Also need to show distributivity.)

*2*Vector spaces¶

Definition: A set $V$ equipped with the operations of addition and scalar multiplication over a field $\mathbb F$ for which the following properties hold:

- Commutativity of addition
- Associativity of addition
- Existence of an additive identity
- Existence of additive inverses
- Existence of a multiplicative identity (in $\mathbb F$ - left and right multiplication)
- Distributivity: $a(u+v) = au +av$ for $a \in \mathbb F$ and $u, v \in V$, as well as $(a+b)u = au + bu$ for $a, b \in \mathbb F$ and $u \in V$.

*2.1*Examples of vector spaces¶

- $\mathbb R^2$. An element of this vector space is, well, a vector, starting at the origin. To prove that this is a vector space, we define addition and scalar multiplication in the usual way, then verify the properties.
- $\mathbb F^n$ in general, over $\mathbb F$.
- Polynomials. $x \mapsto a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$.
- Matrices. $M_{m\times n}(\mathbb F) = $ the set of $m\times n$ matrices with entries in $\mathbb F$. We define addition only on matrices with the same dimensions.

Exercises: Verify that the following are vector spaces: $\mathbb R^n$ over $\mathbb R$, $\mathbb C^n$ over $\mathbb C$, $\mathbb C^n$ over $\mathbb R$. Verify that the following is not a vector space: $\mathbb R^n$ over $\mathbb C$.