Tuesday, January 8, 2013

Complex numbers and vector spaces

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

1Complex 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.)

2Vector 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:

1. Commutativity of addition
2. Associativity of addition
3. Existence of an additive identity
4. Existence of additive inverses
5. Existence of a multiplicative identity (in $\mathbb F$ - left and right multiplication)
6. 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 $u \in V$.

2.1Examples of vector spaces¶

1. $\mathbb R^2$. 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.
2. $\mathbb F^n$ in general, over $\mathbb F$.
3. Polynomials. $x \mapsto a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$.
4. 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 $\mathbb C$.