vector space $\mathbb{R}$, namely the $(x,y)$ coordinate plane, shown below.
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
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}
## Properties of a vector space
You might be familiar with the concept that one can perform a number of **operations** betweenvectors. Some important operations that are relevant in this course are are:
-**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,
The property that a vector space is complete upon scalar multiplication and vector addition is
The property that a vector space is complete upon scalar multiplication and vector addition is
also known as the {\bf closure condition}.
also known as the **closure condition**.
\item There exists a {\bf null element} $\vec{0}$ such that $\vec{a}+\vec{0} =\vec{0}+\vec{a}=\vec{a} $.
- There exists a **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
-**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
element of the same vector space, $-\vec{a}$, such that their addition results
in the null element, $\vec{a} + \lp -\vec{a}\rp = \vec{0}$.
in the null element, $\vec{a} + \lp -\vec{a}\rp = \vec{0}$.
%
This element it called the inverse element.
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}
Some definitions:
- For a complex number $z = a + b {{\rm i}}$, $a$ is called the *real
part*, and $b$ the *imaginary part*.
- The *complex conjugate* $z^*$ of $z = a + b {{\rm i}}$ is defined as
$$z^* = a - b{{\rm i}},$$ i.e., taking the complex conjugate means
flipping the sign of the imaginary part.
### Addition
For two complex numbers, $z_1 = a_1 + b_1 {{\rm i}}$ and
$z_2 = a_2 + b_2 {{\rm i}}$, the sum $w = z_1 + z_2$ is given as
Complex numbers can be rendered on a two-dimensional (2D) plane, the
*complex plane*. This plane is spanned by two unit vectors, one
horizontal, which represents the real number 1, whereas the vertical
unit vector represents ${\rm i}$.
Note that the norm of $z$ is the length of this vector.
#### Addition in the complex plane
Adding two numbers in the complex plane corresponds to adding the
horizontal and vertical components:
We see that the sum is found as the diagonal of a parallelogram spanned
- 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.
by the two numbers.
## Complex functions
- There are other properties, both for what we are interested in these are sufficient.
The last expression is called the *Euler identity*.
**Exercise** Check that this function obeys
$$\exp(z_1) \exp(z_2) = \exp(z_1 + z_2).$$ You need sum- and difference
formulas of cosine and sine.
### The polar form
A complex number can be represented by two real numbers, $a$ and $b$
which represent the real and imaginary part of the complex number. An
alternative representation is a *vector* in the complex plane, whose
horizontal component is the real, and vertical component the imaginary
part. However, it is also possible to characterize that vector by its
*length* and *direction*, where the latter can be represented by the
angle the vector makes with the horizontal axis:
The angle with the horizontal axis is denoted by $\varphi$, just as in
the case of polar coordinates. In the context of complex numbers, this
angle is denoted as the *argument*. We have:
> A complex number can be represented either by its real and imaginary
> part, corresponding to the Cartesian coordinates in the complex plane,
> or by its *norm* and its *argument*, corresponding to polar
> coordinates. The norm is the length of the vector, and the argument is
> the angle it makes with the horizontal axis.
From our previous discussion on polar coordinates we can conclude that
for a complex number $z = a + b {\rm i}$, its real and imaginary parts
can be expressed as $$a = |z| \cos\varphi$$ $$b = |z| \sin\varphi$$ The
inverse equations are $$|z| = \sqrt{a^2 + b^2}$$
$$\varphi = \arctan(b/a)$$ for $a>0$. In general:
$$\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
\pi + \arctan(b/a) & {\rm for ~} a<0{\rm~and~}b>0;\\
-\pi + \arctan(b/a) &{\rm for ~} a<0 {\rm ~ and ~} b<0.
\end{cases}$$
It turns out that using this magnitude $|z|$ and phase $\varphi$, we can write any complex number as
$$z = |z| e^{{\rm i} \varphi}$$
When increasing $\varphi$ with $2 \pi$, we make a full circle and reach the same point on the complex plane. In other words, when adding $2 \pi$ to our argument, we get the same complex number!
As a result, the argument $\varphi$ is defined up to $2 \pi$, and we are free to make any choice we like, such as
Some useful values of the complex exponential to know by heart are $e^{2{\rm i } \pi} = 1 $, $e^{{\rm i} \pi} = -1 $ and $e^{{\rm i} \pi/2} = {\rm i}$.
From the first expression, it also follows that
$$e^{{\rm i} (y + 2\pi n)} = e^{{\rm i}\pi} {\rm ~ for ~} n \in \mathbb{Z}$$
As a result, $y$ is only defined up to $2\pi$.
Furthermore, we can define the sine and cosine in terms of complex exponentials:
We see that during multiplication, the norm of the new number is the *product* of the norms of the multiplied numbers, and its argument is the *sum* of the arguments of the multiplied numbers. In the complex plane, this looks as follows:
**Example** Find all solutions solving $z^4 = 1$.
Of course, we know that $z = \pm 1$ are two solutions, but which other solutions are possible? We take a systematic approach:
We only consider differentiation and integration over *real* variables. We can then regard the complex ${\rm i}$ as another constant, and use our usual differentiation and integration rules:
