Fixing figure, adding admonitions

src/3_vector_spaces.md

@@ -41,7 +41,7 @@ We will denote by ${\mathcal V}^n$ the **vector space** composed by all possible
 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 
+!!! info "Vector basis" 
     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.
@@ -51,22 +51,26 @@ and we can write a generic vector  $\vec{v}$  as
 
 $$\vec{v} = (v_1, v_2, \ldots, v_n) = v_1 \vec{a}_1 + v_2 \vec{a}_2 + \ldots v_n \vec{a}_n \, .$$
 
-However, one could choose another different basis, denoted by $\vec{b}_1,\vec{b}_2,\ldots,\vec{b_n}$, where the same vector would be expressed in terms of a different set of components
+However, one could choose a different basis, denoted by $\vec{b}_1,\vec{b}_2,\ldots,\vec{b_n}$, where the same vector would be expressed in terms of a different set of components
 
 $$ \vec{v} = (v'_1, v'_2, \ldots, v'_n) = v'_1 \vec{b}_1 + v'_2 \vec{b}_2 + \ldots v'_n \vec{b}_n \, .$$
 
-so while the vector remains the same, the values of its components depends on the specific choice of basis.
+Thus, while the vector remains the same, the values of its components depend on the specific choice of basis.
 
-The most common basis is the **Cartesian basis**, where for example for $n=3$ one has
+The most common basis is the **Cartesian basis**, where for example for $n=3$:
 
 $$\vec{a}_1 = (1, 0, 0) \, ,\qquad \vec{a}_2 = (0, 1, 0)\, ,\qquad \vec{a}_3 = (0, 0, 1) \, .$$
   
-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.
+!!! warning ""
+    The elements of a vector basis must be **linearly independent** from one another, meaning
+    that none of them can be expressed as a linear combination of the other basis vectors.
 
 We can consider one example in the two-dimensional real vector space $\mathbb{R}$, namely the $(x,y)$ coordinate plane, shown below.
 
-![image](figures/3_vector_spaces_1.jpg)
+<figure markdown>
+  ![image](figures/3_vector_spaces_1.jpg)
+  <figcaption></figcaption>
+</figure>
   
 We see how the same vector $\vec{v}$ can be expressed in two different bases. 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.