Let $f,g: V \rightarrow W$ be linear operators. Prove that $r(f + g) \leq r(f) + f(g)$   Note: r = rank  My idea was to use matrix representation and to prove that the rank of matrix C ($C= F+G$) can't be bigger than the sum of $r(F) + r(G)$,but to use that idea I must prove that $r(f) = r(g)$ which I don't have an idea how to do. Other idea was to prove that $(f+g)(v) = f(v) + g(v)$ but also with no luck.
Let A,B be in $M_{mxn}(\mathbb{R})$  Prove that if rank(A + B) = rank(A) + rank(B), then col(A) ∩ col(B) = {0}  I started with a proof by contradiction, since we know that rank(A + B) ≤ rank(A) + rank(B). I assume that if rank(A + B) &lt; rank(A) + rank(B), then col(A) ∩ col(B) = {0}. I realize that we don't have to do the > case. However I am a little confused on how to show col from rank.
Google Sheets QUERY won't display cell text if other cells have numbers I'm not sure what is happening in my Google Sheet query and could really use some help. I have  that demonstrates and explains the issue.   Basically, I'm using a query to pull data from a Google Spreadsheet that is populated by a Google Form. If the form submits only numbers (100), there is no problem. However, if someone submits a string ($100 or 100 dollars), the query will not display the string. It will only display form submissions that are 100% numbers.   Also, if the 1st submission is a string (not 100% numbers) the query will work.   Here is my query  =query(data!B2:CL,"select B, C, D, E, F, G, H, I, J, CI, BZ, CA, BA  where J = 'Industry Certification'",false)   Any ideas on why this is happening?  Also, I'm aware that I can limit the form to numbers only. I'm trying to avoid that and learn 'why' this is happening. What does role db_owner allow I have been trying to troubleshoot a login that is not able to view certain tables in a SQL Server 2012 database. In doing so I find I don't quite understand what membership in the db_owner role allows. I can understand the other roles such as db_datareader and db_datawriter but I remain confused as to what db_owner allows. Java Generics: List, List, List Suggestions for a batch audio converter? Any suggestions for a good, free audio converter for Mac?  I used to use Max.app but it has long become abandonware and is now super buggy and crashy. I just need something I drop a couple (or 20 thousand) Apple Lossless files into and let it crank away making me some AAC compressed files or some MP3 or whatever. Ok, I just bought a brand new original iPad. Sounds impossible, right? Well, we just opened the box, and it is not even charging. There is no indication that this iPad even works even though it is technically "new." Is this something that Apple will need to fix or do I just need to let it sit on the charger overnight and hope it wakes up from a 2 year sleep in its box? Adding a comment should not make me fail the review audit I just failed these   in First Post, I read the question and thought it seemed reasonable but the user was deleted so to be on the safe side I clicked add comment and I failed the audit.  I'm not sure why adding a comment would be considered a negative action, is the correct option to click No Action Needed or edit?  What was I supposed to do in this situation?  Adding a comment should not be a review audit fail. Find the value of $x,y$ for the equation $12x+5y=7$ using number theory . I tried the solution using other ways like:  $12x+5y=7$  $12x=7-5y$  $x=7-5y/12$  putting the value of $x$  $12(7-5y/12)+5y=7$  both $12$ will cancel out  $7-5y+5y=7$ here comes the problem please help me ? Today 4/30/20 my seemingly worthless USO call option increased in value.   Specifically: USO1 $2.5 Call 5/8  On 4/29 it was valued at $.04 and stayed there.   4/30 I checked after hours and now it’s up to $.08? Which was wasn’t even the highest it went to today.  So who is buying this contract?  Who is allowed to buy this contract? Bc I’m only able to sell and from what I understood, nobody is able to buy it.   I have screenshots and will post if requested. Thank you What is the most efficient way to deep clone an object in JavaScript? What is the most efficient way to clone a JavaScript object? I've seen obj = eval(uneval(o)); being used, but . I've done things like obj = JSON.parse(JSON.stringify(o)); but question the efficiency.  I've also seen recursive copying functions with various flaws.  I'm surprised no canonical solution exists. Possible Duplicate:          I was wondering if there's any good reason to still use blob fields in a database. A couple of years ago I worked with a DB with a bunch of images in it, the DB was very slow and I couldn't see any good reason to keep the images inside the DB, so I got the images out and stored the filenames instead.  Was this a smart move? What would you do in my place? Why does magnet attract iron but not other metals? If $\omega \in \Omega^q(M)$, and $(\omega|_{U})_p = 0$, is $\omega_p = 0$? If I have a manifold $M$ and $\omega \in \Omega^q(M)$, considered an open subset $U$ of $M$, and choose a point $p \in M$, is it true that $(\omega|_{U})_p=0 \rightarrow \omega_p=0$? With $\omega|_{U}$ I mean the pull-back of $\omega$ by the embedding $F:U\to M$. The converse is obviously true, but this implication? Select from list of reasons when performing downvotes When typesetting an article documentclass, the sections appear as bold in the table of contents, with no leading dots. I would like to change these to non-bold, and insert some leading dots. I tried the following, but it seems to be ignored. Any ideas?  % change section headers and page numbers to normal instead of bold font \renewcommand{\cftsecfont}{ \normalfont } \renewcommand{\cftsecpagefont}{% \normalfont } % use leader dots with section headers \renewcommand{\cftsecleader}{\cftdotfill{\cftdotsep}} Is there a streaming API for JSON? How to escape '$' and '#' in Facelets/EL? Listing multiple compound words Evaluating $ \lim_{x \rightarrow 0}\left (\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$ Syntax error due to using a reserved word as a table or column name in MySQL How to call C++ function from C?