Jason
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 Apr22 comment How do I begin proving this binomial coefficient identity: ${n\choose 0} - {n\choose 1} + {n\choose 2} - {n\choose 3} + \dots = 0$ That was the answer at the back of the book. But I don't get what just that statement means. Is that a proof to just "say" that? May15 comment Basic independent probability question Okay, according to the definition there (and Google), so it is times, which means it's $^$. Thanks. May15 comment Basic independent probability question Hmm...the probability the purse won't hold them all is the probability that all three coins come up heads (as you mentioned). But what is that, in math? Okay well it can't be $1/2 + 1/2 + 1/2$ because that probability would be $> 1$. So is it really $1/2 * 1/2 * 1/2$? Isn't there something about adding independent events and multiplying dependent events? Or are these events dependent? Or have I got it hopelessly wrong? Apr26 comment Does one always use augmented matrices to solve systems of linear equations? Wow, great additions, thanks. Apr26 comment Are these two permutation matrices equivalent? Thanks. 8 more chars to go Apr26 comment Does one always use augmented matrices to solve systems of linear equations? Awesome, a confirmation. Thanks again. Apr26 comment Does one always use augmented matrices to solve systems of linear equations? Thank you! So sorry for the obvious messy question. Apr26 comment Does one always use augmented matrices to solve systems of linear equations? Updated the question, sorry for all the ambiguity. Apr26 comment Does one always use augmented matrices to solve systems of linear equations? @HenrySwanson, great, so if you're provided a $A$ and a $B$, you must augment $B$ as $[A|b]$ to properly solve the matrix right? This is just basic linear algebra, as we've just covered basic concepts, without any fancy mathwork. Apr26 comment Does one always use augmented matrices to solve systems of linear equations? So basically my question is, if you were forced to use matrices instead of easier equation subtraction/elimination (from grade school), must you augment the $3$ and $6$ (i.e. $[A|b]$ to properly solve the matrix? Otherwise you're kind of not performing EROs to each side, if you only solve $[A]$ right? My question may seen stupidly obvious, but I guess I'm confused... Apr26 comment Does one always use augmented matrices to solve systems of linear equations? Oh, I guess I meant, specific to linear algebra homework/test problems where you're given $Ax = b$ and told to solve it using matrix elimination. Sorry! Apr26 comment Does one always use augmented matrices to solve systems of linear equations? I'm afraid I still don't understand. Can't you perform row operations on just $[A]$ instead of $[A|b]$? Apr26 comment Does one always use augmented matrices to solve systems of linear equations? Why not? Are you saying you could rearrange the equations as $x + 0y - 3 = 0$ and $-x + y - 6 = 0$? Jan25 comment Order of operations of multiple Matrix Elementary Row Operations I was finally able to reproduce this correct solution. Thank you for all your help. Jan24 comment Order of operations of multiple Matrix Elementary Row Operations Also, I think I'm not understanding $E_{13}E_{31}$ anymore. What is the order of multiplying those two matrices on a matrix $A$? Is it $E_{13} * E_{31} * A$? Or $E_{31} * E_{13} * A$? Or $A * E_{13} * E_{31}$? Or $A * E_{31} * E_{13}$? Jan24 comment Order of operations of multiple Matrix Elementary Row Operations What is "P(say)"? By the way, I've uploaded images of the book's question and answer. Jan24 comment Order of operations of multiple Matrix Elementary Row Operations Just kidding, it says "Test on the identity matrix!" after that. But does that mean anything? Jan24 comment Order of operations of multiple Matrix Elementary Row Operations The line that starts with "The book's answer is", the words and matrix after that is exactly pixel for pixel what the solutions has. No other letters before or after. Jan24 comment Order of operations of multiple Matrix Elementary Row Operations You mean multiplying $E_{13}E_{31}$ (my answer) with some random 3 by 3 matrix $M$? Jan24 comment Order of operations of multiple Matrix Elementary Row Operations Yeah, something like that. I added the (adds row 1 to row 3) to point out that the subscripts are reversed. Like, $E_{xy}$ means first y and then x.