Update 3_vector_spaces.md

src/3_vector_spaces.md

@@ -128,25 +128,19 @@ by a general scalar $\lambda$, the result is another vector which also belongs
 
 ## 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.
+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.
 
-First of all, let us remind ourselves how we express vectors in the standard Euclidean space.
-In two dimensions, the position of a point $\vec{r}$
-when making explicit the Cartesian basis vectors reads
+First of all, let us remind ourselves how we express vectors in the standard Euclidean space. In two dimensions, the position of a point $\vec{r}$ when making explicit the Cartesian basis vectors reads
 $$
-\vec{r}=x\textbf{\hat{i}}+y\textbf{\hat{j}} \, .
+\vec{r}=x \hat{i}+y\hat{j} \, .
 $$
-As mentioned above, the  unit vectors $\textbf{\hat{i}}$ and $\textbf{\hat{i}}$
-form an *orthonormal basis* of this vector space, and we call $x$ and $y$ the *components* of $\vec{r}$ with respect to the directions spanned by the basis vectors.
+As mentioned above, the  unit vectors $\hat{i}$ and $\hat{j}$ form an *orthonormal basis* of this vector space, and we call $x$ and $y$ the *components* of $\vec{r}$ with respect to the directions spanned by the basis vectors.
 
-Recall also that the  choice of basis vectors is not unique, we can use any other pair of orthonormal unit vectors $\textbf{\^i'}$ and $\textbf{\^j'}$, and express the vector $\vec{r}$ in terms of these new basis vectors as
+Recall also that the  choice of basis vectors is not unique, we can use any other pair of orthonormal unit vectors $\hat{i}$ and $\hat{j}$, and express the vector $\vec{r}$ in terms of these new basis vectors as
 $$
-    \vec{r}=x'\textbf{\^i'}+y'\textbf{\^j'}=x\textbf{\^i}+y\textbf{\^j} \, ,
+\vec{r}=x'\textbf{\^i'}+y'\textbf{\^j'}=x\textbf{\^i}+y\textbf{\^j} \, ,
 $$
-with $x'\neq x$ and $y'\neq y$. So while the vector itself does not depend on the basis, the values of its components
-are basis dependent.
+with $x'\neq x$ and $y'\neq y$. So while the vector itself does not depend on the basis, the values of its components are basis dependent.
 
 We can also express the vector $\vec{r}$ in the following form
 $$