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1500 questions
74
votes
6 answers
How do I tell if matrices are similar?
I have two $2\times 2$ matrices, $A$ and $B$, with the same determinant. I want to know if they are similar or not.
I solved this by using a matrix called $S$:
$$\left(\begin{array}{cc}
a& b\\
c& d
\end{array}\right)$$
and its inverse in terms…
user4681
- 805
74
votes
1 answer
What is the logic/rationale behind the vector cross product?
I don't think I ever understood the rationale behind this.
I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from the cosine rule. (Do correct me if I'm…
Danxe
- 1,695
74
votes
2 answers
Is it mathematically valid to separate variables in a differential equation?
I read the following statement in a book on Calculus, as part of my mathematics course:
Technically this separation of $\frac{dy}{dx}$ is not mathematically valid. However, the resulting integration leads to correct answer.
The book also contains…
Devarsh Ruparelia
- 1,120
74
votes
14 answers
How to stop forgetting proofs - for a first course in Real Analysis?
I am taking my first course in analysis. I like the subject. I study it almost on a daily basis. I try to prove theorems on my own without even looking at the hints. If I really get stuck I just read the first line of the proof and then try to…
MAS
- 1,921
73
votes
2 answers
Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few):
Characteristic polynomial of a matrix of $1$'s
Eigenvalues of the rank one matrix…
Marc van Leeuwen
- 115,048
73
votes
17 answers
What is a real world application of polynomial factoring?
The wife and I are sitting here on a Saturday night doing some algebra homework. We're factoring polynomials and had the same thought at the same time: when will we use this?
I feel a bit silly because it always bugged me when people asked that in…
Dan
- 871
73
votes
3 answers
Mathematical research of Pokémon
In competitive Pokémon-play, two players pick a team of six Pokémon out of the 718 available. These are picked independently, that is, player $A$ is unaware of player $B$'s choice of Pokémon. Some online servers let the players see the opponents…
Andrew Thompson
- 4,431
73
votes
4 answers
Check if a point is within an ellipse
I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.
How do I determine if a point $(x,y)$ is within the area bounded by the ellipse?
73
votes
3 answers
How likely is it not to be anyone's best friend?
A teenage acquaintance of mine lamented:
Every one of my friends is better friends with somebody else.
Thanks to my knowledge of mathematics I could inform her that she's not alone and $e^{-1}\approx 37\%$ of all people could be expected to be in…
hmakholm left over Monica
- 286,031
73
votes
5 answers
Matrix is conjugate to its own transpose
Mariano mentioned somewhere that everyone should prove once in their life that every matrix is conjugate to its transpose.
I spent quite a bit of time on it now, and still could not prove it. At the risk of devaluing myself, might I ask someone else…
George
- 1,837
73
votes
2 answers
Conjecture $_2F_1\left(\frac14,\frac34;\,\frac23;\,\frac13\right)=\frac1{\sqrt{\sqrt{\frac4{\sqrt{2-\sqrt[3]4}}+\sqrt[3]{4}+4}-\sqrt{2-\sqrt[3]4}-2}}$
Using a numerical search on my computer I discovered the following inequality:
$$\left|\,{_2F_1}\left(\frac14,\frac34;\,\frac23;\,\frac13\right)-\rho\,\right|<10^{-20000},\tag1$$
where $\rho$ is the positive root of the polynomial…
HWﾠ
- 1,063
- 7
- 9
73
votes
3 answers
What do prime ideals in $k[x,y]$ look like?
Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like?
As far as I know, the maximal ideals of $k[x,y]$ are of the form $(x-a,y-b)$ where $a,b\in k$. What can we say about the…
user14242
- 2,870
73
votes
10 answers
Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$
How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$?
Edit: note that while…
Isaac
- 36,557
73
votes
8 answers
How can one prove that $e<\pi$?
This question is inspired by another one, asking to prove that something approximately equal to $1.2$ is bigger than something approximately equal to $0.9$. The numerical answer to this question was (expectedly) downvoted, though in my opinion it is…
Start wearing purple
- 53,234
- 13
- 164
- 223
73
votes
12 answers
Why is empty set an open set?
I thought about it for a long time, but I can't come up some good ideas. I think that empty set has no elements,how to use the definition of an open set to prove the proposition.
The definition of an open set: a set S in n-dimensional space is…
python3
- 3,494