For vector spaces $V,W$, probe that $\Lambda (V)\otimes\Lambda (W) \cong \Lambda (V\oplus W)$ For vector spaces $V,W$, probe that $\Lambda (V)\otimes\Lambda (W) \cong \Lambda (V\oplus W)$.  I have tried to use the universal property, but I can not create the necessary linear transformation.
Exterior power "commutes" with direct sum I know that for vector spaces $V, W$ over a field $K$, we have the following identity :  $$ \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \simeq \Lambda^n(V \oplus W) $$ which holds in a canonical way : for $0 \le k \le n$, the maps  $$ \Lambda^k(V) \times \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W), \\ (v_1 \wedge\cdots \wedge v_k,w_{k+1} \wedge \cdots \wedge w_n) \mapsto (v_1 \wedge \cdots \wedge v_k) \wedge (w_{k+1} \wedge \cdots \wedge w_n) $$ factors through the tensor product's universal properties, which gives us a map $\varphi_k : \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \to \Lambda^n(V \oplus W)$, and since the category of $K$-modules has biproducts, we get a canonical map  $$ \varphi = \bigoplus_{k=0}^n \varphi_k : \bigoplus_{k=0}^n \left[ \Lambda^k(V) \otimes_K \Lambda^{n-k}(W) \right] \longrightarrow \Lambda^n(V \oplus W) $$  What if I replace $K$ by a (commutative ring with $1$) ring $A$? Does this map have a kernel? I know it is surjective (it suffices to "look at it", i.e generators of the codomain are obviously all hit by $\varphi$). I don't know if it helps, but I am working in integral domains. In the field case, this is obvious if $V$ and $W$ are finitely generated, since one can use dimension arguments. I wouldn't mind if the above identity held only in the finitely generated case either. If it worked in general it would be cool though.   Note that when trying to compute the inverse map with the universal property, there seems to be a problem... my guess would have been $$ (v_1 + w_1,\cdots,v_n + w_n) \mapsto \bigoplus_{k=0}^n (v_1 \wedge \cdots \wedge v_k) \otimes (w_{k+1} \wedge \cdots \wedge w_n) $$ and this map is obviously $A$-multilinear, but there is a problem with the alternating property ; I get some terms left. I'm guessing this is the wrong map... if there is one.
Browse for Servers does not display local instance I just installed SQL Server and SQL Server Management Studio but it doesn't show anything in servers list and I don't know how to connect to it.  This shows that I have installed SQL Server:        But in this image you can see that my Browse for Servers dialog is empty:        Network servers is empty too. Suppose we have a situation like      A ball of some mass $m$ with some velocity collides with rod hinged at point $A$. Is momentum conserved in this situation? I know that hinge will give impulsive force on rod but that is internal force when we take ball and rod together as a system so that shouldn't stop us using conservation of momentum equation Let $f(n) = \sum_{(k,n)=1} e^\frac{2\pi ik}{n}$. Show that $f(n) = \sum_{k=1}^{n}\left(\sum_{d\mid (k,n)} \mu(d)e^{\frac{2\pi ik}{n}} \right) $ amsalpha-like bibtex style with manual labels I'm sorry if this seems a bit too basic, but I guess I'm just looking to confirm understanding here. I get the sense I'd have to do this in two steps, and I've started trying to grok correlation matrices, but it's just starting to seem really involved. I'm looking for a concise explanation (ideally with hints towards a pseudocode solution) of a good, ideally quick way to generate correlated random numbers.   Given two pseudorandom variables height and weight with known means and variances, and a given correlation, I think I'm basically trying to understand what this second step should look like:     height = gaussianPdf(height.mean, height.variance)    weight = gaussianPdf(correlated_mean(height.mean, correlation_coefficient),                          correlated_variance(height.variance,                          correlation_coefficient))    How do I calculate the correlated mean and variance? But I want to confirm that's really the relevant problem here.  Do I need to resort to matrix manipulation? Or do I have something else very wrong in my basic approach to this problem? 'pyrcc4' is not recognized as an internal or external command? C:\Users\Department\.qgis2\python\plugins\np&gt;pyrcc4 -py2 resources.qrc -o resources.py      'pyrcc4' is not recognized as an internal or external command,      operable program or batch file.   How do I solve this problem? Difference between Ubuntu and Linux Mint Can someone please tell me the differences between Ubuntu and Linux Mint ?   I thought they were almost the same thing until I downloaded Ubuntu 13.10 versions of softwares for my Linux mint 16 system, and half of them didn't work. (I did that as those are the links I got when I searched for Linux mint versions) What is the /boot partition really for? I'm reading a relatively old text on Linux partitions and file systems (the LPIC 1 Certification Bible). It says:     Some versions of the Linux boot loaders cannot access a kernel that is   outside the first 1024 cylinders on a disk. By putting the /boot   partition at the beginning of the drive you can be assured of not   having a problem when accessing the kernel at boot. This problem shows   itself most often in cases of dual booting Linux along with another   operating system that is on the first partition.   Why would a boot loader have "no access to the kernel outside the first 1024 cylinders on a disk"?  Also, what does "putting the /boot partition at the beginning of the drive" mean? Does it make a difference to CP if you feed candies before or after evolving? Convert a list to a data frame Allowing a group Read-Write Access to a directory What iron-clad reason did Dumbledore have for believing Snape was on his side What iron-clad reason did Dumbledore have for believing Snape was on his side? Maybe he was a good actor and a double agent. Was it just that he loved (or pretended to) Lily or he used legilimency? I have two atoms, both of which are either bosons or fermions, with four allowed energy states: $E_1 = 0$, $E_2 = E$, $E_3 = 2E$, with degeneracies 1, 1, 2 respectively.  What's the difference between the partition functions of a pair of two bosons and that of a pair of two fermions? What is a rocket engine thrusting against in space? I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? Are there Ubuntu security notices feeds for specific releases I need help with this exercise.  I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$  I think I should use some convergence theorem, but I'm stuck.  Thanks a lot! Using restroom on airplane while fasten seatbelt light is on? This question about  over on  made me wonder what a passenger is supposed to do when they urgently have to use the restroom while the fasten seat belt light is on. FAA regulations require passengers to obey the seat-belt sign, even if flight attendants do not.  I saw this happen on a flight where an older gentleman asked the flight attendant if he could use the restroom, and she said "not until the light is off". The flight had been free of turbulence for 15 minutes or so by this time, and the flight attendants were walking freely throughout the cabin, collecting post-meal trash, etc. The gentleman who asked to use the restroom appeared to be in quite some discomfort, and practically ran to the restroom when the light went off 10 minutes later.  So what is a passenger supposed to do when he isn't permitted to use the lavatory, when he has to go and can't wait? Just go in their seat? Indexing redundantly named vector in R How do I keep my character centred on screen? Hi I want to try Ubuntu 20.04 without installing it due to some driver issues in my laptop. But I already have 2 OS ( Windows 10 and Ubuntu 20.04 [Dualboot] ), so will trying Ubuntu mess with the already installed Operating systems? And I know how to make live usb and booting from it but don't know how to exit it and then bring my laptop back to normal. Tried finding procedure on internet but reliable results are not found.