Prove the following results for the eigenvalues of an $n \times  n$ matrix $A$:   (a) $0$ is an eigenvalue for $A$ if and only if $A$ is not invertible.  (b) $A$ and $A^T$ have the same eigenvalues.
Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$.
How to disprove this fallacy that derivatives of $x^2$ and $x+x+x+\dots\quad(x\text{ times})$ are not same. Possible Duplicate:          \begin{align*} x^2 &amp;= \underbrace{x + x + x + \dots + x}_{x \text{ times}}, \\ \therefore \frac{\mathrm{d}}{\mathrm{d}x} (x^2) &amp;= \frac{\mathrm{d}}{\mathrm{d}x} (\underbrace{x + x + x + \dots + x}_{x \text{ times}}) \\ &amp;= \underbrace{1 + 1 + 1 + \dots + 1}_{x \text{ times}} \\ &amp;= x. \end{align*}  But we know that $$ \frac{\mathrm{d}}{\mathrm{d}x} (x^2) = 2x. $$  So what is the problem?  My take is that we cannot differentiate both sides because $\underbrace{{x+x+x+\cdots+x}}_{x \text{  times}}$ is not fixed and thus $1$ is not equal to $2$. This is my follow up question. My previous question is here:   Problem:  If I add the image in my table, I get the following output as shown in the image below. Why is the Co-Supervisor going below?   I am using this code:  \begin{tabular}{|l|c|r|} \hline \bf Author: &amp; \hspace{5pt} &amp; \bf Research Supervisor:\\ \hline Mr. X &amp;  \hspace{5pt} &amp; Prof. Dr. Z\\ \hline \includegraphics[width=0.40\textwidth,height=10mm]{Figures/images} &amp;  \hspace{5pt} &amp; \bf Co-Supervisor:\\ \hline &amp;  \hspace{5pt} &amp; Prof. Dr. Y\\ \hline \end{tabular}   What I want is shown below:   PS: Please ignore the table borders, they are for demo only, I won't be using them in my final code.  The final product: that I desire is: My friend installed Windows 7 on his computer (upgrading from XP).  He wasn't sure if his processor supported x64 or not, so he installed the x86 version, just to be on the safe side.  After I looked it up, it turns out his processor does support x64, and now we want to upgrade.  The question is: is it possible to upgrade without completely re-installing the OS?  Can I? Or do I have to format the C: partition again? What is the use of percent signs (%) at the end of lines? (Why is my macro creating extra space?) Locate the ".Rprofile" file generating default options I wonder if it is possible to consider $\mathbb R$ as a Cartesian product $X\times X$ for some set $X$. From the point of view of the dimensionality, there are spaces with a Hausdorff dimension $1/2$ (sort of Cantor sets), but I guess there are other problems in this construction  unrelated to the dimension.  Edited: replying on the comment by Asaf. I want that for all $r\in\mathbb R$ there exists a unique representation $r=\langle x,y\rangle$ where $x\in X$ and $y\in X$. Also, what if we want to have $X$ to be a topological space and $h:X\times X\to\mathbb R$ to be a homeomorphism? Restore Unity Greeter I've installed XFCE Desktop environment, but when I decide to switch back to Unity, while uninstalling the XFCE packages, the LightDM keep starts like XFCE, and I want to restore the Unity one. I think that is called Unity Greeter. Can someone help me? How can I run scripts automatically when Ubuntu starts up so I don't have to run them manually after startup? What kind of power management circuit do I need to embedded a rechargeable battery in my electronics project? I am doing a project where my electronics runs on 5V which I wanted to power from a 7.2v lipo regulated down to that voltage. I'd really like to include the opportunity to charge the lipo without removing it from the project. What kind of circuit/components do I need to do it properly?  Say, would putting a battery charging ic at the battery's terminals in parallel to my system be enough to achieve what I need? Why does the product topology allow proper subsets for only finitely many elements? I'm trying to make a list of ways to tell if a given degree sequence is impossible. For example $3,1,1$  is not possible because there are only 3 vertices in total so one can't have degree 3.  The list so far   vertices has degree equal to or larger than number of vertices sum of degrees is odd for n vertices if one has degree n-1 and another has degree 0 for n vertices the sum of the degrees cannot be greater than $n(n-1)$ because this would be have more edges than a complete graph How do I format a number in Java? I am trying to pickle some mangoes the Indian way. I boiled the jar in hot water, dried it and then added the mixture of spices, salt and mangoes to the jars after the mixture had cooled. I made the mixture in a frying pan to try and remove as much water as I could. After it was cooled, I put it in the jar and heated some mustard oil till it just barely started smoking. I let that cool and poured it over the mango spice mixture.  For the first three days it was fine. On the fourth day, I opened it to test the taste and ended up adding more salt. Now, there seems to be gas forming in the jar. I've noticed it a few times now. When I open the jar, I hear a whooshing noise as air escapes. It's quite a lot, since I opened the container about 3 times in a 10 hour period and each time gas escaped. Has this pickle gone bad? Or is this something that's normal? Intersection of finitely generated ideals Let $I$, $J$ be finitely generated ideals in a ring $A$ (commutative with identity). I know that the intersection need not be finitely generated: can somebody give me an example?   Thanks. Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$ A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called even if $f(-x) = f(x)$ $ \forall x \in \mathbb{R}  $   A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called odd if $f(-x) = -f(x)$ $ \forall x \in \mathbb{R}  $   Let $U_{e}$ denote the set of real-valued even functions on $\mathbb{R}$ and let $U_{o}$ denote the set of real valued odd functions on $\mathbb{R}$.   Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$   This question from Q24 of "Linear Algebra Done Right" 3rd Edition - Chp 1.4 (Subspaces). The author of the book does minimal work with functions during the chapter. I have no idea how to approach the question. In fact, of these, only Charcoal shows a recent time stamp. What's the canonical way to check for type in Python? What is the best way to check whether a given object is of a given type? How about checking whether the object inherits from a given type?  Let's say I have an object o. How do I check whether it's a str? Modelling the number of apples an orchard yields with diminishing returns per tree This is   in the University of Washington's precal book.  Sylvia has an apple orchard. One season, her 100 trees yielded 140 apples per tree. She wants to increase her production by adding more trees to the orchard. However, she knows that for every 10 additional trees she plants, she will lose 4 apples per tree (i.e., the yield per tree will decrease by 4 apples). How many trees should she have in the orchard to maximize her production of apples?  Instructing a precal class, I walked through my approach to answering this in class (without preparing) and ended up doing something strange that may have confused the students. How would you clearly explain a way to get the function for the number of apples that we are trying to maximize? More generally, how would approach answering this question? It'd be nice to collect a some different approaches to demonstrate to my students that there is not a single rigid correct way to solve a problem. Down Arrow Key is not working Duplication formula for gamma function Using the  for $\Gamma(x)$ and $\Gamma\Big(x + \frac12\Big)$, how can I prove the ? This is problem $10.7.3$ in the book , by Boros and Moll.  Any help is highly appreciated.