From 27ba59628474396d1fc74aa553bde8e30666e6b5 Mon Sep 17 00:00:00 2001
From: Maciej Topyla <m.m.topyla@student.tudelft.nl>
Date: Sun, 4 Sep 2022 11:52:06 +0000
Subject: [PATCH] Update src/3_vector_spaces.md
---
src/3_vector_spaces.md | 58 ++++++++++++++++++++++--------------------
1 file changed, 30 insertions(+), 28 deletions(-)
diff --git a/src/3_vector_spaces.md b/src/3_vector_spaces.md
index 4424858..b1fb080 100644
--- a/src/3_vector_spaces.md
+++ b/src/3_vector_spaces.md
@@ -97,7 +97,7 @@ You might be already familiar with the concept of performing a number of various
### Vector products
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 vectors.
+There are two types of vector products; where the end result is a scalar (so just a number) and where the end result is another vector.
!!! info "Scalar product of vectors"
The scalar product of vectors is given by $$ \vec{a}\cdot \vec{b} = a_1b_1 + a_2b_2 + \ldots + a_nb_n \, .$$
@@ -112,39 +112,41 @@ There are two types of vector productions, one where the end result is a scalar
Note that this cross-product can only be defined in *three-dimensional vector spaces*. The resulting vector
$\vec{c}=\vec{a}\times \vec{b} $ will have as components $c_1 = a_2b_3-a_3b_2$, $c_2= a_3b_1 - a_1b_3$, and $c_3= a_1b_2 - a_2b_1$.
-- A special vector is the **unit vector**, which has a norm of 1 *by definition*. A unit vector is often denoted with a hat, rather than an arrow ($\hat{i}$ instead of $\vec{i}$). To find the unit vector in the direction of an arbitrary vector $\vec{v}$, we divide by the norm:
-$$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$
+### Unit vector and orthogonality
-- Two vectors are said to be **orthonormal** of they are perpendicular (orthogonal) *and* both are unit vectors.
+!!! info "Unit vector"
+ A special vector is the **unit vector**, which has a norm of 1 *by definition*. A unit vector is often denoted with a hat, rather than an arrow ($\hat{i}$ instead of $\vec{i}$). To find the unit vector in the direction of an arbitrary vector $\vec{v}$, we divide by the norm: $$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$
-Now we are ready to define in a more formal way what are vector spaces,
+!!! info "Orthogonality"
+ Two vectors are said to be **orthonormal** of they are perpendicular (orthogonal) *and* both are unit vectors.
+
+Now we are ready to define in a more formal way what vector spaces are,
an essential concept for the description of quantum mechanics.
The main properties of **vector spaces** are the following:
-- A vector space is **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
-
-$$\vec{a}, \vec{b} \in {\mathcal V}^n, \qquad \vec{c} = (\vec{a} + \vec{b})
-\in {\mathcal V}^n \, ,\qquad \forall\,\, \vec{a}, \vec{b} \,.$$
-
-- A vector space is **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
-$$\vec{a} \in {\mathcal V}^n, \qquad \vec{c} = \lambda \vec{a}
-\in {\mathcal V}^n \qquad \forall\,\, \vec{a},\lambda \, .$$
-The property that a vector space is complete upon scalar multiplication and vector addition is
-also known as the **closure condition**.
-
-- There exists a **null element** $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
-
-- **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, $\vec{a} + ( -\vec{a}) = \vec{0}$. This element it called the **inverse element**.
+!!! ""
+ A vector space is **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
+ $$\vec{a}, \vec{b} \in {\mathcal V}^n, \qquad \vec{c} = (\vec{a} + \vec{b}) \in {\mathcal V}^n \, ,\qquad \forall\,\, \vec{a}, \vec{b} \,.$$
+
+!!! ""
+ A vector space is **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 $$\vec{a} \in {\mathcal V}^n, \qquad \vec{c} = \lambda \vec{a}
+ \in {\mathcal V}^n \qquad \forall\,\, \vec{a},\lambda \, .$$
+ The property that a vector space is complete upon scalar multiplication and vector addition is
+ also known as the **closure condition**.
+
+!!! ""
+ There exists a **null element** $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
+
+!!! ""
+ **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, $\vec{a} + ( -\vec{a}) = \vec{0}$. This element it called the **inverse element**.
A vector space comes often equipped with various multiplication operations between vectors, such as the scalar product mentioned above
(also known as *inner product*), but also other operations such as the vector product or the tensor product. There are other properties, both for what we are interested in these are sufficient.
--
GitLab