Update src/3_vector_spaces.md

src/3_vector_spaces.md

@@ -2,28 +2,33 @@
 title: Vector Spaces
 ---
 
-# Vector spaces
+# 3. Vector spaces
 
 The lecture on vector spaces consists of **three parts**:
 
-- [Definition and basis dependence](#definition-and-basis-dependence)
+- [3.1. Definition and basis dependence](#31-definition-and-basis-dependence)
 
-- [Properties of a vector space](#properties-vector-space)
+- [3.2. Properties of a vector space](#32-properties-vector-space)
 
-- [Matrix representation of vectors](#matrix-representation-vectors)
+- [3.3. Matrix representation of vectors](#33-matrix-representation-vectors)
 
-and at the end of the lecture one can find the corresponding [Problems](#problems)
+and at the end of this lecture note, there is a set of corresponding exercises
+
+- [3.4 Problems](#34-problems)
+
+---
 
 The contents of this lecture are summarised in the following **videos**:
 
-- [3_vector_spaces_video1](https://www.dropbox.com/s/evytrbb55fgrcze/linear_algebra_01.mov?dl=0)
+1. [Vector spaces: Introduction](https://www.dropbox.com/s/evytrbb55fgrcze/linear_algebra_01.mov?dl=0)
 
-- [3_vector_spaces_video2](https://www.dropbox.com/s/1530xb7zbuhwu6u/linear_algebra_02.mov?dl=0)
+2. [Operations in vector spaces](https://www.dropbox.com/s/1530xb7zbuhwu6u/linear_algebra_02.mov?dl=0)
 
-- [3_vector_spaces_video3](https://www.dropbox.com/s/5lwkxd8lw5uwri9/linear_algebra_03.mov?dl=0)
+3. [Properties of vector spaces](https://www.dropbox.com/s/5lwkxd8lw5uwri9/linear_algebra_03.mov?dl=0)
 
+**Total video lentgh: ~16 minutes**
 
-## Definition and basis dependence
+## 3.1. Definition and basis dependence
 
 A vector $\vec{v}$ is a mathematical object characterised by both a **magnitude** and a **direction**, that is, an orientation in a given space.
   
@@ -34,10 +39,13 @@ $$\vec{v} = (v_1, v_2,\ldots, v_n) \, ,$$
 We will denote by ${\mathcal V}^n$ the **vector space** composed by all possible vectors of the above form.
 
 The components of a vector, $\{ v_i\}$ can be **real numbers** or **complex numbers**,
-depending on whether we have a real or a complex vector space. Note that 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
-of a vector space.
-  
+depending on whether we have a real or a complex vector space. 
+
+!!! note 
+    Note that the above expression of $\vec{v}$ in terms of its components assume that we are using a 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 $\vec{a}_1,\vec{a}_2,\ldots,\vec{a_n}$,
 and we can write a generic vector  $\vec{v}$  as
 
@@ -66,7 +74,7 @@ though the magnitude and direction of the vector itself remain unchanged.
 For many problems, both in mathematics and in physics, the appropiate choice of the vector space basis will significantly facilitate
 its solution.
     
-## Properties of a vector space
+## 3.2. Properties of a vector space
 
 You might be familiar with the concept that one can perform a number of **operations** between vectors. Some important operations that are relevant in  this course are are:
 
@@ -140,7 +148,7 @@ A vector space comes often equipped with various multiplication operations betwe
 (also known as *inner product*), but also  other operations such as the vector product or the tensor product. There are other properties, both for what we are interested in these are sufficient.
 
 
-## Matrix representation of vectors
+## 3.3. Matrix representation of vectors
 
 It is advantageous to represent vectors with a notation suitable for matrix manipulation and operations. As we will show in the next lectures, the operations involving states in quantum systems can be expressed in the language of linear algebra.
 
@@ -199,7 +207,7 @@ Therefore, we see that the scalar product of vectors in Euclidean space can be e
 
 ***
 
-## Problems
+## 3.4. Problems
 
 **1)** [:grinning:] Find a unit vector parallel to the sum of $\vec{r}_1$ and $\vec{r}_2$, where we have defined
 $$\vec{r}_1=2\vec{i}+4\vec{j}-5\vec{k} \, , $$ and $$\vec{r}_2=\vec{i}+2\vec{j}+3\vec{k} \, .$$.