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# Linear Algebra Basics

Some knowledge about linear algebra it’s assumed, in particular notions of:

• Vector Operation
• Geometrical Representation
• Function properties
• Vector Functions
• Matrix Algebra

Optionaly, knowing concepts such as dual vector, forms, and tensors will make this learning experince more interesting.

#### Getting Started

Recalling the set of the real numbers $\mathbb{R}$, and the set opration called Cartesian product, we can define new sets such as $\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}$, which is a new set of tupples, that we call R 2 or $\mathbb{R}^{2}$, where if we take any member of $\mathbb{R}^2$ it takes the form of $(x_1, x_2)$ where $x_1, x_2 \in \mathbb{R}$.

NOTE: In general if we take the cartisian product of $\mathbb{R}$ with itself $n$-times, we’ll get $\mathbb{R}^{n}$.

If we look at a member or $\mathbb{R}^n$ we can express it as ${x} = (x_1, \dots, x_n)$, so we can start building up from here.

First we can define the operations between memebers of $\mathbb{R}^n$, which are:

We define a function $\_ + \_: \R^n + \R^n \to \R^n$ with the following properties
1. $\forall x, y \in \R^n, x + y = y + x$
1. $\forall x, y, z \in \R^n, x + (y + z) = (x + y) + z$
1. $\exists! 0_{\R^n} \in \R^n$ such that $\forall x \in \R^n, x + 0_{\R^n} = x$
1. $\forall x \in \R^n, \exists! y \in \R^n$ such that $x + y = 0_{\R^n}$
Scalar Multiplication
Given $x \in \mathbb{R}^n$ and $c \in \mathbb{R}$, we define $c x = (cx_1, \dots, cx_n)$.
Inner Product
We say that a function $\langle \_, \_ \rangle: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ , is an inner product if it’s linear under boths entries, it commutes, and its positive defined
1. $\forall x, y \in \mathbb{R}^n, \langle x, y \rangle = \langle y, x \rangle$
1. $\forall x, y, z \in \mathbb{R}^n$ and $\forall a \in \mathbb{R}$, $\langle ax + y, z \rangle = a\langle x, z \rangle + \langle y, z \rangle$
1. $\forall x \in \R$ ,$\langle x, x \rangle \geq 0$ and $\langle x, x \rangle = 0 \iff x = 0_{\mathbb{R}^n}$
Norm
We say that a function $\left\lVert \_ \right\rVert : \mathbb{R}^n \to \mathbb{R}$ is a norm if the following is true
1. $\forall x \in \mathbb{R}^n, \left\lVert x \right\rVert \geq 0$ and $\left\lVert x \right\rVert = 0 \iff x = 0_{\mathbb{R}^n}$
1. $\forall x \in \mathbb{R}^n$ and $\forall c \in \mathbb{R}$, $\left\lVert cx \right\rVert = |c| \left\lVert x \right\rVert$
1. $\forall x, y \in \mathbb{R}^n, \left\lVert x + y \right\rVert \leq \left\lVert x \right\rVert + \left\lVert y \right\rVert$
Metric
We say that a function $d: \mathbb{R}^n \to \mathbb{R}$ is a metric if:
1. $\forall x, y \in \R^n, d(x, y) \geq 0$ and $d(x, y) = 0 \iff x = y$
1. $\forall x, y \in \R^n, d(x, y) = d(y,x)$
1. $\forall x, y, z \in \R^n, d(x, y) \leq d(x,z) + d(y,z)$
Euclidian Vectors

$(x_1 + y_1, \dots, x_n + y_n)$.