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+---
+title: Vector Spaces
+---
+
+# Complex numbers
+
+The lecture on vector spaces consists of two parts, each with their own video:
+
+- [Definition and basis dependence](#definition-and-basis-dependence)
+- [Properties of a vector space](#properties-vector-space)
+
+**Total video length: xxx **
+
+## Definition and basis dependence
+
+<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.
+
+
+
+
+
+
+
+
+
+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
+$$w = w_1 + w_2 {{\rm i}}= (a_1 + a_2) + (b_1 + b_2) {{\rm i}}$$ where
+the parentheses in the rightmost expression have been added to group the
+real and the imaginary part. A consequence of this definition is that
+the sum of a complex number and its complex conjugate is real:
+$$z + z^* = a + b {{\rm i}}+ a - b {{\rm i}}= 2a,$$ i.e., this results
+in twice the real part of $z$. Similarly, subtracting $z^*$ from $z$
+yields $$z - z^* = a + b {{\rm i}} - a + b {{\rm i}}= 2b{\rm i},$$ i.e.,
+twice the imaginary part of $z$ (times $\rm i$).
+
+### Multiplication
+
+
+For the same two complex numbers $z_1$ and $z_2$ as above, their product
+is calculated as
+$$w = z_1 z_2 = (a_1 + b_1 {{\rm i}}) (a_2 + b_2 {{\rm i}}) = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1) {{\rm i}},$$
+where the parentheses have again be used to indicate the real and
+imaginary parts.
+
+A consequence of this definition is that the product of a complex number
+$z = a + b {{\rm i}}$ with its conjugate is real:
+$$z z^* = (a+b{{\rm i}})(a-b{{\rm i}}) = a^2 + b^2.$$ The square root of
+this number is the *norm* $|z|$ of $z$:
+$$|z| = \sqrt{z z^*} = \sqrt{a^2 + b^2}.$$
+
+### Division
+
+The quotient $z_1/z_2$ of two complex numbers $z_1$ and $z_2$ as above,
+can be evaluated by multiplying the numerator and denominator by the
+complex conjugate of $z_2$:
+$$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
+Check this!
+
+**Example**
+$$\begin{align}
+\frac{1 + 2{\rm i}}{1 - 2{\rm i}} &= \frac{(1 + 2{\rm i})(1 + 2{\rm i})}{1^2 + 2^2} = \frac{1+8{\rm i} -4}{5}\\
+&= -\frac{3}{5} + {\rm i} \frac{8}{5}
+\end{align}$$
+
+### Visualization: the complex plane
+
+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
+by the two numbers.
+
+## Complex functions
+
+<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/7XtR_wDSqRc" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+
+
+Real functions can (most of the times) be written in terms of a Taylor series:
+$$f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (x-x_{0})^{n}$$
+We can write something similar for complex functions,
+when replacing the *real* variable $x$ with its *complex* counterpart $z$:
+$$f(z) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(x_{0})}{n!} (z-x_{0})^{n}$$
+
+For this course, the most important function is the *complex exponential function*, at which we will have a look below.
+
+### The complex exponential function
+The complex exponential is used *extremely often*.
+It occurs in Fourier transforms and it is very convenient for doing calculations
+involving cosines and sines.
+It also makes doing many common operations on complex number a lot easier.
+
+The exponential function $f(z) = \exp(z) = e^z$ is defined as:
+$$\exp(z) = e^{x + {\rm i}y} = e^{x} + e^{{\rm i} y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
+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
+$$\begin{align}
+-\pi < \varphi < \pi \textrm{ (left)} \\
+-\frac{\pi}{2} < \varphi < \frac{3 \pi}{2} \textrm{ (right)} \end{align} $$
+
+
+
+
+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:
+$$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
+$$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2}$$
+
+Most operations on complex numbers are easiest when converting the complex number to its *polar form*, using the exponential.
+Some operations which are common in real analysis are then easily derived for their complex counterparts:
+$$z^{n} = \left(r e^{{\rm i} \varphi}\right)^{n} = r^{n} e^{{\rm i} n \varphi}$$
+$$\sqrt[n]{z} = \sqrt[n]{r e^{{\rm i} \varphi} } = \sqrt[n]{r} e^{{\rm i}\varphi/n} $$
+$$\log(z) = log \left(r e^{{\rm i} \varphi}\right) = log(r) + {\rm i} \varphi$$
+$$z_{1}z_{2} = r_{1} e^{{\rm i} \varphi_{1}} r_{2} e^{{\rm i} \varphi_{2}} = r_{1} r_{2} e^{{\rm i} (\varphi_{1} + \varphi_{2})}$$
+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:
+$$\begin{align} z = e^{{\rm i} \varphi} & \Rightarrow z^4 = e^{4{\rm i} \varphi} = 1 \\
+& \Leftrightarrow 4 \varphi = n 2 \pi \\
+& \Leftrightarrow \varphi = 0, \varphi = \frac{\pi}{2}, \varphi = -\frac{\pi}{2}, \varphi = \pi \\
+& \Leftrightarrow z = 1, z = i, z = -i, z = -1 \end{align}$$
+
+## Differentiation and integration
+
+<iframe width="100%" height=315 src="https://www.youtube-nocookie.com/embed/JyftSqmmVdU" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
+
+
+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:
+$$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
+$$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
+
+## Bonus: the complex exponential function and trigonometry
+
+Let us show some tricks where the simple properties of the exponential
+function helps in re-deriving trigonometric identities.
+
+1. Take $|z_1| = |z_2| = 1$, and $\arg{(z_1)} = \varphi_1$ and
+ $\arg{(z_2)} = \varphi_2$. Then it is easy to see that
+ $z_i = \exp({\rm i} \varphi_i)$, $i=1, 2$. Then:
+ $$z_1 z_2 = \exp[{\rm i} (\varphi_1 + \varphi_2)].$$ The left hand
+ side can be written as
+ $$\begin{align}
+ z_1 z_2 & = \left[ \cos(\varphi_1) + {\rm i} \sin(\varphi_1) \right] \left[ \cos(\varphi_2) + {\rm i} \sin(\varphi_2) \right] \\
+ & = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2 + {\rm i} \left( \cos\varphi_1 \sin\varphi_2 +
+ \sin\varphi_1 \cos\varphi_2 \right).
+ \end{align}$$
+ On the other hand, the right
+ hand side can be written as
+ $$\exp[{\rm i} (\varphi_1 + \varphi_2)] = \cos(\varphi_1 + \varphi_2) + {\rm i} \sin(\varphi_1 + \varphi_2).$$
+ Comparing the two expressions, equating their real and imaginary
+ parts, we find
+ $$\cos(\varphi_1 + \varphi_2) = \cos\varphi_1 \cos\varphi_2 - \sin\varphi_1 \sin\varphi_2;$$
+ $$\sin(\varphi_1 + \varphi_2) = \cos\varphi_1 \sin\varphi_2 +
+ \sin\varphi_1 \cos\varphi_2.$$ Note that we used the resulting
+ formulas already in order to derive the properties of the
+ exponential function. The point is that you can use the properties
+ of the complex exponential to quickly find the form of gonometric
+ formulas which you easily forget.
+
+2. As a final example, consider what we can learn from the derivative
+ of the exponential function:
+ $$\frac{d}{d\varphi} \exp({\rm i} \varphi) = {\rm i} \exp({\rm i} \varphi) .$$
+ Writing out the exponential in terms of cosine and sine, we see that
+ $$\cos'\varphi + {\rm i} \sin'\varphi = {\rm i} \cos\varphi - \sin\varphi.$$
+ where the prime $'$ denotes the derivative as usual. Equating real
+ and imaginary parts leads to $$\cos'\varphi = - \sin\varphi;$$
+ $$\sin'\varphi = \cos\varphi.$$
+
+## Summary
+
+- A complex number $z$ has the form $$z = a + b \rm i$$ where $a$ and
+ $b$ are both real, and $\rm i^2 = 1$. The real number $a$ is called
+ the *real part* of $z$ and $b$ is the *imaginary part*. Two complex
+ numbers can be added, subtracted and multiplied straightforwardly.
+ The quotient of two complex numbers $z_1=a_1 + \rm i b_1$ and
+ $z_2=a_2 + \rm i b_2$ is
+ $$\frac{z_1}{z_2} = \frac{z_1 z_2^*}{z_2 z_2^*} = \frac{(a_1 a_2 + b_1 b_2) + (-a_1 b_2 + a_2 b_1) {{\rm i}}}{a_2^2 + b_2^2}.$$
+
+- Complex numbers can also be characterised by their *norm*
+ $|z|=\sqrt{a^2+b^2}$ and *argument* $\varphi$. These coordinates
+ correspond to polar coordinates in the complex plane. 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 $$|z| = \sqrt{a^2 + b^2}$$
+ $$\varphi = \begin{cases} \arctan(b/a) &{\rm for ~} a>0; \\
+ \pi + \arctan(b/a) & {\rm for ~} a<0 {\rm ~ and ~} b>0;\\
+ -\pi + \arctan(b/a) &{\rm ~ for ~} a<0 {\rm ~ and ~} b<0.
+ \end{cases}$$
+ The complex number itself then becomes
+ $$z = |z| e^{{\rm i} \varphi}$$
+
+- The most important complex function for us is the complex exponential function, which simplifies many operations on complex numbers
+ $$\exp(z) = e^{x + {\rm i}y} = e^{x} \left( \cos y + {\rm i} \sin y\right).$$
+ where $y$ is defined up to $2 \pi$.
+ The $\sin$ and $\cos$ can be rewritten in terms of this complex exponential as
+ $$\cos(x) = \frac{e^{{\rm i} x} + e^{-{\rm i} x}}{2}$$
+ $$\sin(x) = \frac{e^{{\rm i} x} - e^{-{\rm i} x}}{2}$$
+ Because we only consider *differentiation* and *integration* over *real variables*, the usual rules apply:
+ $$\frac{d}{d\varphi} e^{{\rm i} \varphi} = e^{{\rm i} \varphi} \frac{d}{d\varphi} ({\rm i} \varphi) ={\rm i} e^{{\rm i} \varphi} .$$
+ $$\int_{0}^{\pi} e^{{\rm i} \varphi} = \frac{1}{{\rm i}} \left[ e^{{\rm i} \varphi} \right]_{0}^{\pi} = -{\rm i}(-1 -1) = 2 {\rm i}$$
+
+## Problems
+
+1. [:grinning:] Given $a=1+2\rm i$ and $b=-3+4\rm i$, calculate and draw in the
+ complex plane the numbers $a+b$, $ab$, and $b/a$.
+
+2. [:grinning:] Evaluate (a) $\rm i^{1/4}$, (b)
+ $\left(1+\rm i \sqrt{3}\right)^{1/2}$, (c) $\exp(2\rm i^3)$.
+
+3. [:grinning:] Find the three 3rd roots of $1$ and ${\rm i}$ (i.e. all possible solutions to the equations $x^3 = 1$ and $x^3 = {\rm i}$, respectively).
+
+4. [:grinning:] (a) Find the real and imaginary part of
+ $$ \frac{1+ {\rm i}}{2+3{\rm i}}$$
+ (b) Evaluate $$\left| \frac{a+b\rm i}{a-b\rm i} \right|$$
+ for real $a$ and $b$.
+
+5. [:sweat:] For any given complex number $z$, we can take the inverse $\frac{1}{z}$. Visualize taking the inverse in the complex plane. What geomtric operation does taking the inverse correspond to? (Hint: first consider what geometric operation $\frac{1}{z^*}$ corresponds to.)
+
+6. [:grinning:] Compute (a)
+ $$\frac{d}{dt} e^{{\rm i} (kx-\omega t)},$$
+ and (b) calculate the real part of
+ $$\int_0^\infty e^{-\gamma t +\rm i \omega t} dt$$($k$, $x$, $\omega$, $t$ and
+ $\gamma$ are real; $\gamma$ is positive).
+
+7. [:smirk:] Compute
+ $$\int_{0}^{\pi}\cos(x)\sin(2x)dx$$
+ by making use of the Euler identity.
diff --git a/src/vectors_spaces.md b/src/vectors_spaces.md
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-test