I want to prove the following lemma:  Let $B$ be a basis for $V$ and let $T_B: B \rightarrow W$ be a map. Then there exists a unique linear map $T_V: V \rightarrow W$ which extends $f,$ that is, such that $T_V(b)=T_B(b)$ for all $b \in B$  My attempt:  Any vector $v\in V$ can be written as a linear combination of the basis elements: $$v=\sum_{i}^{n} \alpha_{i} b_{i}, \quad b_{i} \in B$$  Where $\alpha_i$ are the coordinates of $v$ with respect to $B$. We can therefore write the transformation of $v$ under $T_V$ as: $$T_{V}(v)=T_{V}\left(\sum_{i}^{n} \alpha_{i} b_{i}\right)=\sum_{i}^{n} \alpha_{i} T_{V}\left(b_{i}\right)$$  Where in the last step I exploited the assumed linearity of A. This means that if $T_V(b_i)=T_B(b_i)\ \forall\ b_i\in B$ then the map $T_B$ has been 'extended' to $T_V$. To prove that $T_V$ is unique we consider another map: $T^{'}_V:V\rightarrow W$ defined over the basis elements $T_V^{'}(b_i)=T_V(b_i)=f(b_i)\ \forall\ b_i \in B$: $$ T_{V}^{'}(v)=\sum_{i}^{n} \alpha_{i} T^{'}_{V}\left(b_{i}\right)=T_V(v) $$ Is this sufficient to prove the Lemma? Is something missing/unclear/incorrect in my intend of a proof?
My professor has stated the following lemma:  Let $U$ and $V$ be two vector spaces and $B$ a basis for $U$. If $f:B\rightarrow V$ is a map, then there exists a unique linear map $A: U \rightarrow V$ such that $A(x)=f(x)\ \forall\ x\in B$  How can I prove this lemma? (This is not a homework problem.)
Show that if $f$ is integrable on $[a, b]$ then $\lvert f \rvert$ is also integrable. Hint: Show that $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ I have: $$U (P , \lvert f \rvert) \ge U (P , f )$$ and $$ L(P , \lvert f \rvert) \ge L(P,f)$$ Thus, $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ Apache 2.4 - 403 HTTP status code How to move an element into another element? I am aware it is common practice to standardize the features for ridge and lasso regression, however, would it ever be more practical to normalize the features on a (0,1) scale as an alternative to z-score standardization for these regression methods? Opportunity attack when moving in and out of range of a creature in the same move? Let's say I'm 10ft away from an enemy. In my turn, I choose to move 5ft to get close enough to attack, I do attack, then I use my remaining movement to move out of range of the enemy. Does this still allow them an opportunity attack? How do I delete folders and files that were older than 7 days on the command line? I want to delete folders and its files were created longer than 7 days using the command line. Table of Variables Filling cells with sequential numbers in an Excel (2003) macro I need to fill an excel column with a sequential series, in this case from -500 to 1000.  I've got a macro to do it, but it takes a lot of lines for something that seems like it should be a single function [something like FillRange(A2:A1502, -500, 1000, 1)].  But if that function exists, I can't find it.  Is the following as simple and elegant as it gets?  'Draw X axis scale Cells(1, 1).Value = "mV" Cells(2, 1).Value = -500 Cells(3, 1).Value = -499 Cells(4, 1).Value = -498  Dim selection1 As Range, selection2 As Range  Set selection1 = Sheet1.Range("A2:A4") Set selection2 = Sheet1.Range("A2:A1502")  selection1.AutoFill Destination:=selection2 Can I use mirroring to make a copy of my Database? What's wrong with this query: INSERT INTO Users( weight, desiredWeight ) VALUES ( 160, 145 ) WHERE id = 1;  It works without the WHERE clause. I've seemed to have forgot my SQL. Can't see a glass object through another glass object in Eevee I was just testing Eevee and found that you cannot view a glass object through another glass. Is there any workaround? It works fine only If I rotate the glass window or put my camera to the back side of the scene On equations $m^2+1=5^n$ How to convert mediawiki syntax to latex? Pressing Enter should not refresh the page FTP and Apache permission issues Prove that $f_n$ converges uniformly on $\mathbb{R}$ Why is the spoken German in many US films and TV shows so inaccurate? Is there a reason why in so many English-language movies and TV series, whenever someone is speaking "German", they (actors, writers etc.) just don't care about if what they say is right or wrong? This can range from:    just minor grammatical errors e.g. "Schieß dem Fenster" in "Die Hard":                                rambling some made-up words and claiming them to be German e.g. "Lebenslangeschicksalssatz" in "How I met your mother"                                wrong German paired with a bad accent e.g. in "Scrubs"                                up to native German speakers talking complete nonsense e.g. Heidi Klum in HIMYM                                  I don't know if this is the same with other languages (for example there is a scene in Scrubs where the cast speaks Spanish, but there might be similar mistakes:                                 Do the creators and actors just not want to speak correct German and at least use correct German words, or is there simply no money in a $100,000,000 movie production to ask a native speaker to look over your script and maybe correct some of the lines? Purely "algebraic" proof of Young's Inequality states that if $a, b \geq 0$, $p, q &gt; 0$, and $\frac{1}{p} + \frac{1}{q} = 1$, then $$ab\leq \frac{a^p}{p} + \frac{b^q}{q}$$ (with equality only when $a^p = b^q$).  Back when I was in my first course in real analysis, I was assigned this as homework, but I couldn't figure it out.  I kept trying to manipulate the expressions algebraically, and I couldn't get anywhere.  But every proof that I've seen since uses calculus in some way to prove this.  For example, a common proof is based on this  and integration.  The proof on Wikipedia uses the fact that $\log$ is concave, which I believe requires the analytic definition of the logarithm to prove (correct me if I'm wrong).  Can this be proven using just algebraic manipulations?  I know that that is a somewhat vague question, because "algebraic" is not well-defined, but I'm not sure how to make it more rigorous. But for example, the proof when $p = q = 2$ is something I would consider to be "purely algebraic":  $$0 \leq (a - b)^2 = a^2 + b^2 - 2ab,$$ so $$ab \leq \frac{a^2}{2} + \frac{b^2}{2}.$$ How can I attach a file to a post? In the chapter on Spellcasting, there is a section called . However, despite the seeming generality of the title, the text of this section only talks about magical effects from spells:     The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.   It makes sense that this section would only talk about spells, since it is in the chapter about spellcasting. However, this seems to leave ambiguous the question of how magical effects from sources other than spells combine. Such an issue could arise, for example, if two different items (or two of the same item) grant a bonus to attack rolls, or a bonus to the same saving throw.  Is there a published rule or official ruling that governs how non-spell magical effects combine?  (This question was inspired by  about the interaction between +X magical ammunition and Oil of Sharpness, and could be considered a generalization of that question.)