I have a 4x3 matrix and want to determine if the columns of my matrix are linearly independent. Is there a way to form a square matrix (4x4) in order get the ability to calculate the determinant and see if they are linearly independent?
We can easily find out if the vectors are dependent or not by finding determinant of matrices formed from the vectors. But if the matrix is not a square matrix, we cannot find the determinant. Then how do we know if the vectors are linearly independent?
Notepad ignoring linebreaks Most documents I'm opening in notepad.exe under windows 7 lately, there are no linebreaks... everything just runs together.  The same documents in any other program, even previewed directly in Explorer, will show the linebreaks correctly.  this is happening with many .info files, .css, .js.. but not all of them.   I am guessing it's some kind of character encoding unix line endings something or other, but wondering if there's anything I can do or if anyone else has experienced similar and if I can make notepad work correctly. I prefer it to notepad++/other text editors for certain uses. How do I reset a sequence in Oracle? In , I can do something like this:  ALTER SEQUENCE serial RESTART WITH 0;   Is there an Oracle equivalent? Do the SO admins have any plans to release the underlying code for SO at all? I appreciate that the code may be part of their competitive advantage, but it strikes me that the underlying system would work for any topic, not just programming.  I am not advocating this, just curious really, but figured it might be under consideration as most of the value in SO is in the questions and answers, which have been released under a relatively open CC license. Javascript add leading zeroes to date Using find with grep, one can locate files that match a pattern:  # find | grep error ./solr-modifiedSolr4/SolrPhpClient/phpdocs/errors.html ./error_log ./includes/classes/error_log   However, using find alone the first file is not found:  # find . -name error* ./error_log ./includes/classes/error_log   Why doesn't find locate the errors.html file when not used with grep? How is find used to show this file as well? Young boy living on Mars discovers that his parents are robots Proving this relation is transitive Let $r$ be a relation on $A \times A$ such that $(a,b) r (c,d) \iff ad = bc.$ How can I show that this relation is transitive, ie. $(a, b)r(c,d)$ and $(c,d)r(e, f) \implies (a,b)r(e,f)$?  I tried to say that $(a,b) r (c,d)$ means that $c=ka$ and $d=kb,$ $k$ a coefficient, such that $ad = bc \iff akb = bka,$ and going on from there, but I'm not sure this is valid for all values of $a, b, c,$ and $d$. How can I show the transitivity of the relation? I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups.  Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then $\operatorname{Aut}(G\times H)\cong\operatorname{Aut}(G)\times \operatorname{Aut}(H)$.  I also know that if $G,H$ are abelian and simple and isomorphic, then $\operatorname{Aut}(G\times H)\cong \operatorname{GL}_2(\mathbf{Z}_p)$ where $p$ is the order of $G$.  What I don't know is the following: what if $G,H$ are simple, yet not abelian? Let us for now focus on a more restricted case where $G\cong H$, in particular, $G=H$. What can we say about the group $\operatorname{Aut}(G\times H)$? (I heard something about this group being related to what is called "wreath product", which I do not know...)  Thanks bunch in advance! Is it "to play a game on someone" or "play games with someone"? I find this expression strange because it's clearly widely used, but seems sort of "unofficial", the "official" version, meaning the one described in dictionaries and grammar books, being playing games with, not on.  Both expressions seem to be current and valid, according to Google Books. I'm just not entirely clear on the context and the usage.  My understanding is that playing games with implies some sort of tricky, mischievous behavior, which may or may not be intentionally malicious, like office politics or standing someone up on a date ("Jane didn't show up...again. Is she playing games with me?"), while playing games on implies a form of studied deception. ("The ad said that the comic book was in mint condition but there are barely noticeable tears... are they trying to play a game on me?") Get text from pressed button This question led to a new package:   I would like to display the ORCID id in an author list in an IEEEtran pdf, like that:  Is there a standard way to include it in an IEEEtran article with PdfLaTeX? I used two approaches from , but I failed. One is to insert the ORCID icon in the author list. I used the blow code: \author{Ali Al-Obaidi \href{https://orcid.org/0000-0000-0000-0000}{\includegraphics[scale=1]{figures/orcid_16x16.png}}}  where orchid_16x16.png is provided in the link . But the result is that the icon is not clear enough, even I replaced it with a different size. The other approach is to use the  package, like that: \usepackage{academicons} \definecolor{orcidlogocol}{HTML}{A6CE39} \author{AAA \href{https://orcid.org/0000-0000-0000-0000}{\textcolor{orcidlogocol}{\aiOrcid}}}  However, the result is like that:  And I also got two errors: &quot;Latex Error: Encoding scheme 'TU' unknown&quot; and &quot;Bad character code (59865)&quot;. In the latest academicons.sty (see ), academicons use the font TU. However, it seems that the compiler PdfLateX cannot support this font. Thus, is there anyone that knows a standard way to include it in an IEEEtran article with PdfLaTeX? Please help me. Uniform convergence for sequences of functions Constant Amortized Time What is meant by "Constant Amortized Time" when talking about time complexity of an algorithm? Get TEXT value of a CLOB OID in Postgresql When is the present perfect tense used instead of the past tense? When is the present perfect tense used instead of the past tense?    I know that the present perfect tense is used when some adverbs (e.g., never, ever) are present in the sentence; the same is true for sentences like the following one.     When you returned, I have been at home since 3:00 PM.   In which other cases should I use the present perfect? Do the following sentences require it?     I have walked downtown everyday for a year.        I have been at home since 3:00 PM. Vertex count appears incorrect after removing doubles I've only just started studying the Blender Python API. I'm writing a script (see below) to automate something I've done by hand : joining two adjacent cubes and cleaning up the resulting mesh.   After joining and removing doubles, I expect to have 12 vertices instead of 16 in the resulting object. In the Info header, this seems to be the case, but when I print to the windows terminal I still see 16. Also I do see the output from remove_double(), here's a screenshot :    So what's wrong here ? What is the right way to access an object's mesh, and count its vertices ?  Joao  import bpy  # Create two adjacent cubes and join them bpy.ops.mesh.primitive_cube_add(location=(1,0,0)) ob = bpy.context.object bpy.ops.mesh.primitive_cube_add(location=(3,0,0)) ob.select = True bpy.ops.object.join()  # Expect 16 vertices, 24 edges, 12 faces result = bpy.context.object print(str(len(result.data.vertices)))  # Go to edit mode, remove double vertices bpy.ops.object.mode_set(mode = 'EDIT') bpy.ops.mesh.remove_doubles()  # Expect 12 vertices, 20 edges, 11 faces result = bpy.context.object print(str(len(result.data.vertices))) Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely.   The hint I was given is to evaluate: $$E[X-Y;X&gt;a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$  which I can write as: $$\int_A(X-Y)dP +\int_B(X-Y)dP$$ where $A=\{X&gt;a, Y\leq a\}$ and $B=\{X\leq a,Y\leq a\}$.  But I need some more hints. Why does ice make such peculiar sounds? I've come across a couple of videos where some interesting sounds are produced using ice.  (Click on the images to see the video.)      Here, they drop a block of ice into a deep crevice and as the block falls, you can hear some strange sounds. These sounds remind me of some sound effects in old cartoons and sci-fi movies. The pitch of the sounds start high and end up low. Is this some kind of doppler effect?      Here, there's a man performing 'nordic skating', making the same sounds as he skates on top of a thin ice sheet. Maybe they are caused by the cracking ice. But why does that sound like this?    There are some other videos too, where they skip stones on frozen lakes.    Here the ice makes some high-frequency sounds which correspond to the chirping of birds.  What are these sounds and why do they sound the way they sound? I am creating table in LaTeX and I need to get the text to perform line break automaticaly when it's too long. And if possible, I'd like to get the text alignment to the left. Is it possible to handle this situation? What is object serialization?