Update 3_vector_spaces.md

src/3_vector_spaces.md

@@ -149,7 +149,7 @@ $$
 which is known as a \emph{column vector}. Note that this notation assumes a specific choice of basis vectors, which is left
 implicit, and displays only the information on its components along this specific basis.
 
-For instance, if we had chosen the basis vectors $\textbf{\^i'}$ and $\textbf{\^j'}$, the components would be $x'$ and $y'$, and the corresponding column vector representing the same vector $\vec{r}$ in such case would be given by
+For instance, if we had chosen another set of basis vectors  $\hat{i}'$ and $\hat{j}'$, the components would be $x'$ and $y'$, and the corresponding column vector representing the same vector $\vec{r}$ in such case would be given by
 $$
 \vec{r}= \begin{pmatrix}x'\\y'\end{pmatrix}.
 $$
@@ -162,11 +162,11 @@ 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*.
 
-The same scalar product can also be expressed in terms of components of $\vec{r_1}$ and $\vec{r_2}$.  When using the $\{\textbf{\^i},\textbf{\^j}\}$ basis, the scalar product will be given by
+The same scalar product can also be expressed in terms of components of $\vec{r_1}$ and $\vec{r_2}$.  When using the $\{ \hat{i}, \hat{j} \}$ basis, the scalar product will be given by
 $$
 \vec{r_1}\cdot\vec{r_2}=x_1\,x_2\,+\,y_1\,y_2 \, .
 $$
-Note that the same result would be obtained if the $\{\textbf{\^i'},\textbf{\^j'}\}$ basis
+Note that the same result would be obtained if the $\{ \hat{i}', \hat{j}' \}$basis
 had been chosen instead
 $$
 \vec{r_1}\cdot\vec{r_2}=x_1'\,x_2'\,+\,y_1'\,y_2' \, .
@@ -183,5 +183,4 @@ where here we say that the vector $\vec{r_1}$ is represented by a *row vector*.
 Therefore, we see that the scalar product of vectors in Euclidean space can be expressed as the matrix multiplication of row and column vectors. The same formalism, as we will see, can be applied for the case of Hilbert spaces in quantum mechanics.
 
 
-
 ## Problems