latex fix

src/3_vector_spaces.md

@@ -82,17 +82,17 @@ solution proces.
 You might be already familiar with the concept of performing a number of various **operations** between vectors, so in this course, let us review some essential operations that are relevant to start working with quantum mechanics:
 
 !!! info "Addition" 
-    I can add two vectors to produce a third vector, $$\vec{a} + \vec{b}= \vec{c}$$.
-    As with scalar addition, also vectors satisfy the commutative property, $$\vec{a} + \vec{b} = \vec{b} + \vec{a}$$.
+    I can add two vectors to produce a third vector, $$\vec{a} + \vec{b}= \vec{c}.$$
+    As with scalar addition, also vectors satisfy the commutative property, $$\vec{a} + \vec{b} = \vec{b} + \vec{a}.$$
     Vector addition can be carried out in terms of their components,
-    $$ \vec{c} = \vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n) =  (c_1, c_2, \ldots, c_n) \, .$$
+    $$ \vec{c} = \vec{a} + \vec{b} = (a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n) =  (c_1, c_2, \ldots, c_n).$$
 
 !!! info "Scalar multiplication" 
-    I can multiply a vector by a scalar number (either real or complex) to produce another vector, $$\vec{c} = \lambda \vec{a}$$. 
+    I can multiply a vector by a scalar number (either real or complex) to produce another vector, $$\vec{c} = \lambda \vec{a}.$$ 
     Addition and scalar multiplication of vectors are both *associative* and *distributive*, so the following relations hold
-    1. $(\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$
-    2. $\lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$
-    3. $(\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$
+    $$1. \vspace{5pt} (\lambda \mu) \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})$$
+    $$2. \vspace{5pt} \lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$$
+    $$3. \vspace{5pt} (\lambda + \mu)\vec{a} = \lambda \vec{a} +\mu \vec{a}$$
 
 !!! info "Vector product" 
     In addition to multiplying a vector by a scalar, as mentioned above, one can also multiply two vectors among them.