Update 3_vectors_spaces.md

src/3_vectors_spaces.md

@@ -2,7 +2,7 @@
 title: Vector Spaces
 ---
 
-# Complex numbers
+# Vector spaces
 
 The lecture on vector spaces consists of two parts, each with their own video:
 
@@ -16,7 +16,184 @@ The lecture on vector spaces consists of two parts, each with their own video:
 <iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/fLMdaMuEp8s" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
 
 A vector $\vec{v}$ is essentially a mathematical object characterised by both
-  a {\bf magnitude} (the length of the vector) and a {\bf direction} (represented by the arrow), that is, an orientation in a given space.
+  a **magnitude** and a **direction**, that is, an orientation in a given space.
+  
+  We can express a vector in terms of its individual **components**.
+  
+  \item Let's assume we have an $n$-dimensional space, meaning that the vector $\vec{v}$ can be oriented
+  in different ways along each of $n$ dimensions.
+  %
+  The expression of $\vec{v}$ in terms of its components is
+  \be
+  \vec{v} = \lp v_1, v_2, \ldots, v_n\rp \, ,
+  \ee
+ 
+  We will denote by ${\mathcal V}^n$ the {\bf vector space} composed
+  by all possible vectors of the above form.
+  %
+  Shortly we will define more precisely what are the mathematical properties of such space.
+
+  The components of a vector, $\{ v_i\}$ can be {\bf real numbers} or {\bf complex numbers},
+  depending on whether we have a real or a complex vector space.
+
+\item The expression above of $\vec{v}$ in terms of its components assume that we are
+  using some specific {\bf basis}.
+%
+  It is important to recall that the same vector can be expressed in terms of different bases.
+
+  A {\bf vector basis} is a set of $n$ vectors that can be used to generate all the elements
+  of a vector space.
+  %
+  For example, a possible basis of  ${\mathcal V}^n$ could be denoted by $\vec{a}_1,\vec{a}_2,\ldots,\vec{a_n}$,
+  and we can write a generic vector  $\vec{v}$  as
+  \be
+  \vec{v} = \lp v_1, v_2, \ldots, v_n\rp = v_1 \vec{a}_1 + v_2 \vec{a}_2 + \ldots v_n \vec{a}_n \, .
+  \ee
+  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
+  \be
+  \vec{v} = \lp v'_1, v'_2, \ldots, v'_n\rp = v'_1 \vec{b}_1 + v'_2 \vec{b}_2 + \ldots v'_n \vec{b}_n \, .
+  \ee
+  so while the vector remains the same, the values of its components depends on the specific choice
+  of basis.
+
+  The most common basis is the {\bf Cartesian basis}, where for example for $n=3$ one has
+  \be
+  \vec{a}_1 = \lp 1, 0, 0 \rp \, ,\qquad \vec{a}_2 = \lp 0, 1, 0 \rp
+  \, ,\qquad \vec{a}_3 = \lp 0, 0, 1 \rp \, ,
+  \ee
+  but other choices of basis are possible.
+  %
+  The elements of a vector basis must be {\bf linearly independent} from each other, meaning
+  that none of them can be expressed as linear combination of the rest of basis vectors.
+
+  \item All this is a bit abstract so let's consider some examples in the two-dimensional real
+    vector space $\mathbb{R}$, namely the $(x,y)$ coordinate plane, shown in Fig.~\ref{fig:L1-R2}.
+    %
+    We see how the same vector $\vec{v}$ can be expressed in two different basis.
+    %
+    In the first one, the Cartesian basis, its components are $\vec{v}=\lp 2,2\rp$.
+    %
+
+    But in the second basis, the components are different, being instead $\vec{v}=\lp 2.4 ,0.8\rp$,
+    though the magnitude and direction of the vector itself remain unchanged.
+    %
+    In the exercises in Brightspace you will find more examples to become familiar with this
+    concept.
+
+  %%%%%%%%%%%%%%%%%%%%%%%%
+\begin{figure}[h]
+  \centering
+  \includegraphics[scale=0.50]{plots/L1-R2.pdf}
+  \caption{\small 
+   The components of the vector $\vec{v}$ depend on the specific basis chosen.
+}  
+\label{fig:L1-R2}
+\end{figure}
+%%%%%%%%%%%%%%%%%%%%%
+
+
+
+
+\item You might be familiar with the concept that one can perform a number of {\bf operations} between
+  vectors. Some important operations that are relevant in this course are are:
+  \begin{itemize}
+
+  \item {\bf 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, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$.
+    %
+    Vector addition can be carried out in terms of their components,
+    \be
+    \vec{c} = \vec{a} + \vec{b} = \lp a_1 + b_1, a_2 + b_2, \ldots, a_n + b_n  \rp = \lp c_1, c_2, \ldots, c_n\rp
+    \ee
+
+  \item  {\bf Scalar multiplication}: 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 {\bf associative} and {\bf distributive}, so the following
+    relations hold
+    \be
+    \lp \lambda \mu\rp \vec{a} = \lambda (\mu \vec{a}) = \mu (\lambda \vec{a})
+    \ee
+     \be
+    \lambda \lp \vec{a} + \vec{b}\rp = \lambda \vec{a} + \lambda \vec{b}
+    \ee
+
+\be
+    \lp \lambda + \mu\rp\vec{a} = \lambda \vec{a} +\mu \vec{a}
+    \ee
+
+  \item {\bf Vector product}: in addition to multiplying a vector by a scalar, as mentioned
+    above, one can also multiply two vectors among them.
+    %
+    There are two types of vector productions, one where the end result is a scalar (so just a number)
+    and the other where the end result is another vector.
+
+    The {\bf scalar production of vectors} is given by
+    \be
+    \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .
+    \ee
+    Note that since the scalar product is just a number, its value will not depend on the specific
+    basis in which we express the vectors: the scalar product is said to be {\bf basis-independent}.
+    
+  \end{itemize}  
+
+
+
+
+ \item Now we are ready to define in a more formal way what are vector spaces,
+   an essential concept for the description of quantum mechanics.
+
+   The main properties of {\bf vector spaces} are the following:
+
+\begin{enumerate}
+
+\item A vector space is {\bf complete upon vector addition}.
+
+  This property means that if
+  two arbitrary vectors  $\vec{a}$ and $\vec{b}$
+  are elements of a given vector space ${\mathcal V}^n$,
+  then their addition should also be an element of the same vector space
+  \be
+  \vec{a}, \vec{b} \in {\mathcal V}, \qquad \vec{c} = \lp \vec{a} + \vec{b}\rp
+  \in {\mathcal V}^n \qquad \forall\,\, \vec{a}, \vec{b}
+  \ee
+
+  \item A vector space is {\bf complete upon scalar multiplication}.
+
+    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
+  \be
+  \vec{a} \in {\mathcal V}, \qquad \vec{c} = \lambda \vec{a}
+  \in {\mathcal V}^n \qquad \forall\,\, \vec{a},\lambda
+  \ee
+
+  The property that a vector space is complete upon scalar multiplication and vector addition is
+  also known as the {\bf closure condition}.
+\item There exists a {\bf null element} $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
+
+\item {\bf Inverse element}: for each vector $\vec{a} \in \mathcal{V}^n$ there exists another
+  element of the same vector space, $-\vec{a}$, such that their addition results
+  in the null element, $\vec{a} + \lp -\vec{a}\rp = \vec{0}$.
+  %
+  This element it called the inverse element.
+
+  \item A vector space comes often equipped with various multiplication operations between vectors, such as the scalar product mentioned above, but also other operations such as the vector product or the tensor product.
+
+\item There are other properties, both for what we are interested in these are sufficient.
+
+\end{enumerate}
+
+\item You will find in Brightspace additional material and examples that you can use to
+  extend your knowledge of linear vector spaces.
+
+\item In the next video we will discuss how to apply these ideas to the case of quantum mechanics.
+
+\end{itemize}