From 4414d80990ba495bbb37a5624a72faa64c46a03b Mon Sep 17 00:00:00 2001
From: Sonia Conesa Boj <s.conesaboj@tudelft.nl>
Date: Sat, 29 Aug 2020 13:49:35 +0000
Subject: [PATCH] Update 3_vector_spaces.md

---
 src/3_vector_spaces.md | 19 ++++---------------
 1 file changed, 4 insertions(+), 15 deletions(-)

diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md
index b85d03f..803ccda 100644
--- a/src/3_vector_spaces.md
+++ b/src/3_vector_spaces.md
@@ -18,9 +18,7 @@ The lecture on vector spaces consists of two parts, each with their own video:
 A vector $\vec{v}$ is essentially a mathematical object characterised by both
 a **magnitude** and a **direction**, that is, an orientation in a given space.
   
-We can express a vector in terms of its individual **components**.
-  
-Let's assume we have an $n$-dimensional space, meaning that the vector $\vec{v}$ can be oriented
+We can express a vector in terms of its individual **components**. Let's assume we have an $n$-dimensional space, meaning that the vector $\vec{v}$ can be oriented
 in different ways along each of $n$ dimensions.
   
 The expression of $\vec{v}$ in terms of its components is
@@ -34,11 +32,7 @@ The components of a vector, $\{ v_i\}$ can be **real numbers** or **complex numb
 depending on whether we have a real or a complex vector space.
 
 The expression above of $\vec{v}$ in terms of its components assume that we are
-using some specific **basis**.
-
-It is important to recall that the same vector can be expressed in terms of different bases.
-
-A **vector basis** is a set of $n$ vectors that can be used to generate all the elements
+using some specific **basis**. It is important to recall that the same vector can be expressed in terms of different bases. A **vector basis** is a set of $n$ vectors that can be used to generate all the elements
 of a vector space.
   
 For example, a possible basis of  ${\mathcal V}^n$ could be denoted by $\vec{a}_1,\vec{a}_2,\ldots,\vec{a_n}$,
@@ -54,7 +48,7 @@ so while the vector remains the same, the values of its components depends on th
 
 The most common basis is the **Cartesian basis**, where for example for $n=3$ one has
 
-$$\vec{a}_1 = (1, 0, 0) \, ,\qquad \vec{a}_2 = (0, 1, 0)\, ,\qquad \vec{a}_3 = (0, 0, 1) \, ,$$
+$$\vec{a}_1 = (1, 0, 0) \, ,\qquad \vec{a}_2 = (0, 1, 0)\, ,\qquad \vec{a}_3 = (0, 0, 1) \, .$$
   
 The elements of a vector basis must be **linearly independent** from each other, meaning
 that none of them can be expressed as linear combination of the rest of basis vectors.
@@ -63,12 +57,7 @@ Let's consider one example in the two-dimensional real vector space $\mathbb{R}$
 
 ![image](figures/3_vector_spaces_1.jpg)
   
-We see how the same vector $\vec{v}$ can be expressed in two different basis.
-    
-In the first one, the Cartesian basis, its components are $\vec{v}=(2,2)$.
-  
-
-But in the second basis, the components are different, being instead $\vec{v}=(2.4 ,0.8)$,
+We see how the same vector $\vec{v}$ can be expressed in two different basis. In the first one (left panel), the Cartesian basis, its components are $\vec{v}=(2,2)$. But in the second basis (right panel), the components are different, being instead $\vec{v}=(2.4 ,0.8)$,
 though the magnitude and direction of the vector itself remain unchanged.
     
 ## Properties of a vector space
-- 
GitLab