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Do I have to recompile my application when I upgrade a third party jar? Idempotent linear operators are projections I'm working on this problem: Suppose $V$ is a Hilbert space and $P : V \to V$ is a linear map such that $P^2= P$ and $\Vert Pf \Vert \leq \Vert f \Vert$ for every $f \in V$. Prove that there exists a closed subspace $U$ of $V$ such that $P = P_U$. I did do some searching online and I know that $P$ is an idempotent matrix and I know that $U$ is supposed to be $\text{range}P$, which I can show is closed. However, I'm having a hard time showing $P_{\text{range}P} = P$. I know that I can decompose $f = g + h$, where $g \in \text{range}P$ and $h$ is in the orthogonal complement of $\text{range}P$. I'm not sure how to proceed. Most of the online searches require that $P$ be self-adjoint, but this question doesn't have this condition. Can I get some help? When should unions be used? Why do we need them? How many rectangles or triangles. Font size setting is missed in iPhone application Can you add a setting for choosing font size in iPhone mobile app?   It's hard to read questions. Lens upgrade paths (sub $1000) for the EF-S 18-55mm IS kit lens for Canon APS-C cameras Determine Which of the following Sets is a Vector Space Determine which of the following sets is a vector space.  $V$ is the line $y=x$, in the $xy$-plane: $V =\{[x,y]: \; y=x\}$  $W$ is the union of the first and second quadrants in the $xy$-plane: $W=\{[x,y]: \; y\geq0\}$  $U$ is the line $y = x+1$ in the $xy$-plane: $U=\{[x,y]: \; y=x+1\}$    I know that the answer is that "$V$" is the only vector space. However how do I come up with a quick proof to show that $V $is a vector space and that the other $2$ aren't? I'm planning on selling a USB external hard drive that currently contains an old Ubuntu installation with stored passwords and banking information.  How can I securely erase the drive before selling it? Can the ship's captain really perform your wedding when in international waters? On , Mark Mayo posted a comment that any marriage has to be on dry land or close to it.  But I'm sure all of us have heard of ships' captains marrying couples at sea.  From the movie Casablanca:     Say, why don't we get married in Marseilles?   ...   Well, let's see. What about the engineer? Why can't he marry us on the train?   ...   Well, why not? The captain on a ship can. It doesn't seem fair that...    So is this fiction after all? If not how does it fit with Mark's comment? What special rules and such should we know?  If I'm travelling at sea with a significant other and wish to tie the knot, can we do so with any captain on any ship on any sea? Must it be international waters? Should I use the past tense with did? Should I use the past tense with did? For example, I was to say:     The important question is: Did they knew what it means or not?   Or should I say:     The important question is: Did they know what it means or not?   In other words: Should I use past tense with did?  I looked at this:    but it didn't really answer my question. Finding an item in a List using C# Find the expansion for $\det(I+\epsilon A)$ where $\epsilon$ is small without using eigenvalue. It seems that I'm missing something about this.  First of all, the series is convergent: $\lim_{n\rightarrow\infty}\frac{2^{-n-1} (n+1)^2}{2^{-n} n^2}=\frac{1}{2}$ (ratio test)  What I tried to do is to find a limit of a partial sum $\lim_{n\rightarrow\infty}S_n$ as follows: $S_n=\frac{\frac{1}{6} n (n+1) (2 n+1)}{\frac{1-\left(\frac{1}{2}\right)^n}{2 \left(1-\frac{1}{2}\right)}}$. Still, the limit is $\infty$ and I'm clearly doing something wrong. I'm a little confused as to when to use  for my physics class. For example, I'm asked to find the average speed of a race car that travels around a circular track with a radius of $500~\mathrm{m}$ in $50~\mathrm{s}$.  Would I need to apply the rules of significant figures to this step of the problem? $$ C = 2\pi (1000~\mathrm{m}) = 6283.19 $$  Or do I just need to apply significant figures at this step? $$ \text{Average speed} = \frac{6283.19~\mathrm{m}}{50~\mathrm{s}} = 125.664~\mathrm{m}/\mathrm{s} $$  Should I round $125.664~\mathrm{m}/\mathrm{s}$  to  $130~\mathrm{m}/\mathrm{s}$ since the number with the least amount of significant figures is two? Proof that all abelian simple groups are cyclic groups of prime order Getting the Volume of an Uneven Mesh Suppose John has 5 sweets. Is there any difference between the following two sentences?     Jack has 3 times as many sweets as John.      Jack has 3 times more sweets than John.   I prefer the first construction and would know unambiguously that Jack has 15 sweets in this case. However in the second construction I would be inclined to think that Jack has 20 sweets, since it seems to suggest 15 sweets in addition to the original 5. Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number Prove: if $|z|=1$ than $\frac{z+1}{z-1}$ is an imaginary number    I have tried to look at $$\frac{z+1}{z-1}=\frac{z+1}{z-1}\cdot \overline{\frac{z-1}{\overline{z-1}}}=\frac{z+1}{z-1}\cdot \frac{\overline{z}-1}{\overline{z}-1}$$  But did not get far Grammatically, what is "It" in the following sentence? I'm currently working at a private academy in Korea, and my boss just asked me a real head-scratcher.   In the sentence:     It doesn't have to be hot and humid for players to lose too much water from their bodies.    I'm aware that "It" is a subject pronoun replacing "The weather." Besides that, however, I'm unsure.   Is it a false subject? I wouldn't think so since it's a pronoun. Is it some crazy kind of indirect object?  Anyway, thank you all for your help! Polynomial passing through two points with specific tangents How can I calculate a Polynomial that passes through the origin and a given point (P) having given tangents at the origin (Ot) and the given point (Op)?  The given point will always be to the right of the origin (Px &gt; 0). The polynomial should be a function of x (i.e. it moves from left to right).  I believe a 3rd degree polynomial will always fit these constraints, but I'm not certain.  I am looking for a function f( x, Px, Py, Ot, Pt ) = ? which plots a polynomial for x, satisfying the four given values.  When x = 0,   f()   = 0 When x = Px,  f()   = Py When x = 0,   f'()  = Ot When x = Px,  f'()  = Pt