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Linear Algebra Basics

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

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

Getting Started

About Vectors

Recalling the set of the real numbers R\mathbb{R}, and the set opration called Cartesian product, we can define new sets such as R×R=R2\mathbb{R} \times \mathbb{R} = \mathbb{R}^{2}, which is a new set of tupples, that we call R 2 or R2\mathbb{R}^{2}, where if we take any member of R2\mathbb{R}^2 it takes the form of (x1,x2)(x_1, x_2) where x1,x2Rx_1, x_2 \in \mathbb{R}.

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

If we look at a member or Rn\mathbb{R}^n we can express it as x=(x1,,xn){x} = (x_1, \dots, x_n), so we can start building up from here.

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

Addition
We define a function _+_:Rn+RnRn\_ + \_: \R^n + \R^n \to \R^n with the following properties
  1. x,yRn,x+y=y+x\forall x, y \in \R^n, x + y = y + x
  1. x,y,zRn,x+(y+z)=(x+y)+z\forall x, y, z \in \R^n, x + (y + z) = (x + y) + z
  1. !0RnRn\exists! 0_{\R^n} \in \R^n such that xRn,x+0Rn=x\forall x \in \R^n, x + 0_{\R^n} = x
  1. xRn,!yRn\forall x \in \R^n, \exists! y \in \R^n such that x+y=0Rnx + y = 0_{\R^n}
Scalar Multiplication
Given xRnx \in \mathbb{R}^n and cRc \in \mathbb{R}, we define cx=(cx1,,cxn)c x = (cx_1, \dots, cx_n).
Inner Product
We say that a function _,_:Rn×RnR\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. x,yRn,x,y=y,x\forall x, y \in \mathbb{R}^n, \langle x, y \rangle = \langle y, x \rangle
  1. x,y,zRn\forall x, y, z \in \mathbb{R}^n and aR\forall a \in \mathbb{R}, ax+y,z=ax,z+y,z\langle ax + y, z \rangle = a\langle x, z \rangle + \langle y, z \rangle
  1. xR\forall x \in \R ,x,x0\langle x, x \rangle \geq 0 and x,x=0    x=0Rn\langle x, x \rangle = 0 \iff x = 0_{\mathbb{R}^n}
Norm
We say that a function _:RnR\left\lVert \_ \right\rVert : \mathbb{R}^n \to \mathbb{R} is a norm if the following is true
  1. xRn,x0\forall x \in \mathbb{R}^n, \left\lVert x \right\rVert \geq 0 and x=0    x=0Rn\left\lVert x \right\rVert = 0 \iff x = 0_{\mathbb{R}^n}
  1. xRn\forall x \in \mathbb{R}^n and cR\forall c \in \mathbb{R}, cx=cx\left\lVert cx \right\rVert = |c| \left\lVert x \right\rVert
  1. x,yRn,x+yx+y\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:RnRd: \mathbb{R}^n \to \mathbb{R} is a metric if:
  1. x,yRn,d(x,y)0\forall x, y \in \R^n, d(x, y) \geq 0 and d(x,y)=0    x=yd(x, y) = 0 \iff x = y
  1. x,yRn,d(x,y)=d(y,x)\forall x, y \in \R^n, d(x, y) = d(y,x)
  1. x,y,zRn,d(x,y)d(x,z)+d(y,z)\forall x, y, z \in \R^n, d(x, y) \leq d(x,z) + d(y,z)
Euclidian Vectors

(x1+y1,,xn+yn)(x_1 + y_1, \dots, x_n + y_n).