2.0 KiB
lecture, date
| lecture | date |
|---|---|
| 4 | 2025-01-16 |
The Inverse Image
The Inverse Image uses a Inverse Function f^{-1} (z) of Z \subset Y written f: x \to y is f^{-1}(z) \equiv \{ x \in X | f(x) \in Z \}.
Complex Numbers
In Complex Numbers, \mathbb{C} = \mathbb{R} \times \mathbb{R} with usual addition of vectors
Multiplying Vectors
Multiply vectors by adding their angles multiplying their lengths.
z = a + i \times b = (a, b)
(b here can be seen as (a, 0) + (0, b) = (a+0, 0+b))
b = (b, 0) \implies i \times b = (0, b)
i \times z rotates z 90\degree counterclockwise.
Proposition
\mathbb{C} is complete (Cauchy Sequence) and Algebraically Complete#For Complex Numbers.
Metric Spaces
Example
Discrete metric on X; d(x, y) = \begin{cases}0, & \text{if}\ x=y\\ 1, & \text{if}\ x \neq y\end{cases}
Vector Spaces
Example
V = R^n = R \times \dots \times R = \{ (x_{1}, \, \dots, x_{n}) | x_{i} \in \mathbb{R} \}
(where the length of R \times \, \dots \times R has n $\mathbb{R}$s.)
V = \mathbb{R}^2 : (x , \, y) + (z, w) \equiv (x + z, \, y + w)
a \times (x, \, y) \equiv (ax, \, ay)
Linear Basis Example
u = 3v_{1} + 5v_{2} + iv_{3}
3 v_{1} \neq 2v_{2}
c_{1}v_{1} + c_{2}v_{2} = 0 \implies c_{1} = 0 = c_{2}
\implies 0 \times v_{1} + 0 \times v_{2} = 0
Continuing the #Vector Spaces#Example but for Linear bases
v_{1} = (1, \, 0, \, 0, \, 0, \, \dots)
v_{2} = (0, \, 1, \, 0, \, 0, \, \dots)
v_{3} = (0, \, 0, \, 1, \, 0, \, \dots)
and the way of writing this would be:
(x_{1}, \, \dots, \, x_{n}) = \Sigma^{n}_{i=1} x_{i} \times v_{i} = 0
\implies x_{i} = 0
Proposition
Any Vector Space V has a Linear Basis, and every basis has the same cardinality referred to as the dim(V) (dimension of V) of V.