1st proof-read complete, ready to merge

src/3_vector_spaces.md

@@ -72,7 +72,7 @@ We can consider one example in the two-dimensional real vector space $\mathbb{R}
   <figcaption></figcaption>
 </figure>
   
-In this figure, you can see how the same vector $\vec{v}$ can be expressed in two different bases. In the first one (left panel), the Cartesian basis is used and its components are $\vec{v}=(2,2)$. But in the second basis (right panel), the components are different, namely $\vec{v}=(2.4 ,0.8)$, while the magnitude and direction of the vector remain unchanged.
+In this figure, you can see how the same vector $\vec{v}$ can be expressed in two different bases. In the first one (left panel), the Cartesian basis is used and its components are $\vec{v}=(2,2)$. In the second basis (right panel), the components are different, namely $\vec{v}=(2.4 ,0.8)$, while the magnitude and direction of the vector remain unchanged.
 
 For many problems, both in mathematics and in physics, the appropiate choice of the vector space basis may significantly simplify the
 solution proces.
@@ -139,8 +139,8 @@ The main properties of **vector spaces** are the following:
     This property means that when I multiply one arbitrary vector  $\vec{a}$,
     element of the vector space ${\mathcal V}^n$, by a general scalar $\lambda$, the result is another vector which also belongs to the same vector space $$\vec{a} \in {\mathcal V}^n, \qquad \vec{c} = \lambda \vec{a}
     \in {\mathcal V}^n \qquad \forall\,\, \vec{a},\lambda \, .$$
-    The property that a vector space is complete upon scalar multiplication and vector addition is
-    also known as the **closure condition**.
+    
+The property that a vector space is complete upon scalar multiplication and vector addition is also known as the **closure condition**.
 
 !!! info ""
     There exists a **null element** $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
@@ -175,7 +175,7 @@ For instance, if we had chosen another set of basis vectors $\hat{i}'$ and $\hat
 $$ \vec{r}= \begin{pmatrix}x'\\y'\end{pmatrix}.$$
 
 We also know that Euclidean space is equipped with a scalar vector product. 
-The scalar product $\vec{r_1}\cdot\vec{r_2}$ of two vectors in 2d Euclidean space is given by
+The scalar product $\vec{r_1}\cdot\vec{r_2}$ of two vectors in 2D Euclidean space is given by
 $$ \vec{r_1}\cdot\vec{r_2}=r_1\,r_2\,\cos\theta \, ,$$
 where $r_1$ and $r_2$ indicate the *magnitude* (length) of the vectors and $\theta$ indicates its relative angle. Note that the scalar product of two vectors is just a number, and thus it must be *independent of the choice of basis*.