Übungen zur Linearen Algebra

Wintersemester 2024

Gruppe 3

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Week 4

We started today with Linda who told us how to decide whether 4 points in $\mathbb{R}^3$ are coplanar. In the second part of the question, she found some collinear points in the plane satisfying a certain condition.
In the second question, we talked about the intersection point of three Seitenhalbierenden in a triangle. Annika showed us that the bisectors are not parallel to each other, find their intersection point and showed that the sum of the OrtVectors equals zero.
For the next one, we had Leonie on the board who talked about different operations on a given set and decided whether these operations define a group structure. We discussed the difference between
- For any $x \in G$, there is an $e \in G$ such that $xe = ex = x$, and
- There is an $e \in G$ such that for any $x \in G$ we have $xe=ex=x$.
Fourth question was a cute one. Dominik L. gave us a proof that if $a^2=e$ for all elements in a group, then the group has to be abelisch.
Then, Sophie told us that if we have a finite group, then for any element $a$, we should have an $n$ such that $a^n=e$.
In the last question, Paul showed us that $F = \{ a+b\sqrt{5} \colon a,b \in \mathbb{Q}\} $ is a field. He then made the following Conjecture: Paul's Conjecture. For every natural number $c$, the set $F = \{ a+b\sqrt{c} \colon a,b \in \mathbb{Q}\} $ is a field. We took a vote of "feelings". If you have free time prove Paul's Conjecture or give a gegenbeispiel.
Some words sound much better in German. Gegenbeispiel is definitely a cooler name than counterexample.
Also sometimes you guys say "vector room" and I think that's funny and cool.
I gave you an optional homework: find alternative names for "basis, linear independence, linearly independent, spanning set, group, abelian group, associativity, field, neutral element". You do not have to do it all, you can also add more. You can submite your answers using this form.

Week 3

Today, I mentioned that we had seminars in our house in Toronto. You can read about them from this link and from this link.
Today, I also mentioned that I wrote about why we denote the square root of $-1$ by $i$. You can read it from this link.
We started this week with Martina. The task wanted us to decide whether or not a given set of vectors form a basis for the given space. We remembered the definition of a basis: a basis is a linearly independent spanning set of vectors. We need to be able to write every vector in our space using these vectors and the operations in our space (addition and scalar multiplication) - this is spanning. And we also need that whenever we represent a vector in terms of vectors in a basis, this representation should be unique - this is linear independence.
We also know that if we have a $d$-dimensional space and if we have exactly $d$ vectors which are linearly independent, then they automatically span. This is what we used in part a and part c of the first question. Checking linear independence was enough to confirm that we have a basis. But in part b, even though the vectors were linearly independent, we did not have a basis of $\mathbb{R}^3$ because they did not span. We were not able to write the vector $$\begin{bmatrix} 0 \\ 0 \\ 1\end{bmatrix}$$ in terms of the given vectors.
In question 2, we had Aaron. I made a joke about this video. The task gave us a line and told us to show that the distance between the origin and any point in this line is at least 1 (and then to find a point on this line whose distance to origin is exactly 1). Aaron argued that if we can find a point $P$ on the line such that our line is orthogonal to the line that contains $OP$, then this would give us the shortest distance. Hence, he proceeded the find the normal vector and showed that this point actually has distance 1 to the origin.
Peter also had a good geometric argument about this question. He said that we can draw the unit circle whose equation is given by $x^2+y^2 = 1$ and we can interpret the question as our line being tangent to the unit circle. So, we can find the intersection point of our line with the circle. The equation for the circle is not a linear equation, it contains squares. So, it is outside of our interest but regardless, this is a nice way of seeing the problem.
In Question 3, we actually proved what Aaron used in the previous question to find the normal vector thanks to Verena. She showed us that the vector $$\mathbf{n} = \begin{bmatrix} a \\ b \end{bmatrix}$$ is orthogonal to the line $a x + by = c$ provided that $(a,b) \neq (0,0)$. Then, she used this fact to show us that we can represent the same line by an infinitely many equations but all of these equations are multiples of each others.
Then, in Question 4, Magdalena volunteered to show us that in dimension 2 and 3, the angle bisector of the angle between two vectors $x$ and $y$ is given by $z = ||x||y + ||y||x$. She did this by computing the angle between $x$ and $z$ and then the angle between $y$ and $z$. On the other hand, at home I had computed the angle between $x$ and $z$ first and then $x$ and $y$ and used the formula $ \cos(2\alpha)=2\cos^2(\alpha)-1$ which is definitely much more annoying.
In Question 5, we had Fabian on the board. First, he argued that if we want to find a vector $z$ that is perpendicular to both $x$ and $y$ in $\mathbb{R}^3$, then we can use the cross product $x \times y$. Then, he used a formula to find the distance between two points. I do not always say this, in fact I very rarely say this, but it might be a good idea to memorise this formula.
We did not have any volunteers for the last question. So, I asked Peter to talk about this. He showed existence and uniqueness of the plane $E$ and concluded his proof with an Ancient Greek version of QED.

Week 2

I taught a linear algebra course at the University of Connecticut a while ago. I was ambitious and started typing up some notes. But I was never able to complete them. So, here is some incomplete notes. They are in English.
Today, we started with Miles who told us about linear independence and how to check whether a given set of vectors is linearly independent or not. The idea was to consider a homogenous vector equation. Such an equation is always consistent: it always has a solution. Namely, you have a solution where you set all the coefficients to zero. Linear independence checks whether there are more solutions. If the zero solution was the only solution, you say that your vectors are linearly independent. If there are more than one solutions, then you say that your vectors are linearly dependent. We discussed how this definition is the same as "you can not write one of the vectors using the remaining vectors and the allowed operations of addition and scalar multiplication". Another thing we mentioned was that it is always good to check with your eyes quickly whether you have some immediate dependence. It may save you time: in part b, it was immediate that the sum of the first two vectors is equal to twice of the third vector. But if you can not find an immediate dependence like this, then you should start with solving your system of linear equations.
For the second question, Annika volunteered. She first observed that if $\lambda = 0$, then the vectors are linearly dependent as the first and the third vectors are the same. So, she made the assumption that $\lambda$ is nonzero. With this assumption, she was allowed to divide by $\lambda$. After performing some row operations, she came to a point where she had to examine two cases: the first case was where $\lambda^2 -2 \neq 0$ and she performed more row operations with this assumption and in the second case she assumed $\lambda^2 - 2=0$ to get a row full of zeroes.
In the third question, Dominik J. checked for us that three vectors in the question form a basis of $\mathbb{R}^3$. He argued that three vectors in $\mathbb{R}^3$ form a basis if and only if they are linearly independent and proceeded to show this. He made some calculation mistakes which we corrected later but if you were taking notes, you should double check these computations. Later, he also showed us how to find the coordinate vector of a given vector with respect to a given basis.
The, Elijas volunteered to talk with us the distinction between linear independence and pairwise linear independence. We learned the term WLOG. ohne Beschränkung der Allgemeinheit! It is a useful term.
In the next question, we tried to make a very precise proof with Laurenz. We proved why it is impossible to have three linearly independent vectors in the span of two linearly independent vectors.
And finally, Paul gave us a proof of an identity involving norms. He also explained us the geometric meaning behind this which was cool.

Week 1

At the beginning of the lecture, Peter asked me my Erdős number and I answered that it is infinity. But today I learned that my Erdős number is actually not that high. I collaborated with Ryo Takahashi. Ryo has more than 10 papers with Shiro Goto who passed away a couple years ago. Shiro Goto worked with David Eisenbud and David Eisenbud collaborated with Persi Diaconis who (apparently) is also a magician. Diaconis himself worked with Erdős on the distribution of the greatest common divisor. So, my Erdős number seems to be 5.
We started the day with a question on two lines in the plane. The two lines were given in different ways and Logy showed us how we can go from one representation of a line to another representation using a very important tool: the normal vector. Then, we discussed what we mean by the intersection of two lines and how we can compute it. We also discussed an alternative method to solve this problem without needing the normal vector.
The second question was about two planes in the three dimensional space this time and the idea was similar to the first question. One must know how to move between two representations of the same plane. Leanka explained us her approach to this problem and helped us compute the intersection.
In the third question, Dominik L. showed us how he handled the system of equations $$\begin{align*} x + cy &= 1 \\ cx + y &= -1 \end{align*}$$ where $c$ is some real number. He used the fact that if two numbers $x$ and $y$ satisfy this system, then they must also satisfy the third equation $(1+c)x + (1+c)y = 0$ which we get by adding the two equations. From here, we realised that for $c=-1$, we have infinitely many solutions. Because, Dominik said, when $(1+c)(x+y)=0$ we either have $1+c = 0$ or $x+y = 0$. If $c = -1$, this equation is satisfied for all $x, y \in \mathbb{R}$. So, our solution set is the line $x-y=1$. If $c \neq -1$, then we know that we must have $x+y = 0$. This gives us $y = -x$. Plugging this into the first equation, we get $x - cx = 1$ and we see that $x = 1/1-c$ provided that $c \neq 1$. When $c=1$, we do not have any solutions. We also thank Peter for helping us with this.
Then, the fourth question had two systems of linear equations and Peter volunteered to solve them. Both of these systems required to see a "trick" to be able to solve them nicely. For the first system, the trick was to add up all the equations. Then, we would get a value for $x_1 + x_2 + x_3 + x_4$. We also know from the question the value of $x_1 + x_2 + x_3$. So, this is enough information to find $x_4$. Same method to find $x_1, x_2$ and $x_3$. For the second question, the trick was to write $2x_i$ as $x_i + x_i$. This gave us $$x_{i} - x_{i-1} = x_{i+1} - x_i$$ for any $2 \leq i \leq 2$ using the equations which are not the top and the bottom equations. Peter called this difference $s$. The top equation told us that $s = x_2 -x_1 = x_1 - a$. Peter then used induction to show that $x_i = a + is$ for any $1 \leq i \leq n$. Finally, combininb this with the last equation, he was able to solve for $s$ in terms of $a$ and $b$ which finished the problem.
In the fifth question, Leon gave us a geometric argument to find 3 points which would form a parallellogram when combined with the three given points in the question (so we found 3 parallellograms). Similarly, in the second part, he found squares.
For the next question, Leonie computed areas of some special triangles inside a rectangle using line equations to find intersection points.
And the last question was solved by Lisa. The point of the question was to be able to appreciate the axioms of a vector space (or properties of addition and scalar multiplication in $\mathbb{R}^2$). The first part was a tedious task using commutativity and associativity multiple times. Lisa proved the second part using two cases: the case where $a$ is the null vector and the case where it is not. But we later discussed that there was no need for that. Finally, in the third part, we saw that if $\alpha a = \alpha b$ then we either have $\alpha = 0$ or $a =b$. Lisa was able to justify each of her steps by referring to a property of the addition (commutativity, associativity, existence of a zero vector, existence of additive inverses) or other facts which were proved in class (such as $0c = o$).