Let T : V → V be a linear transformation of a finite dimensional vector space over a field F to itself. Assume that {$v, T v, T^2v, . . . $} span V for some v ∈ V .   Show that (i) there exists a k such that $v, T v, . . . , T^{k-1}v$ are linearly independent and for some $α_i$ ∈ F  $T^kv = α0v + α1T v + · · · + αk−1T^{k−1}v$  Hence show {$v, T v, T^2v, ...,T^{k-1}v $} is a basis for V.
Let $V$ be an n-dimensional vector space over $\mathbb{R}$ and $T:V \rightarrow V$ be linear. Call a vector $v \in V$ cyclic if $V$ is spanned by $\{v, \ Tv, \ T^2v,..\}$.  Question:  Show that $v$ is cyclic for $T$ then there is some $k \geq 1$ such that   $$B=\{v, \ Tv,..T^{k-1}v \}$$ is linearly independent in $V$ and that $T^kv$ lies in  the span of $B$.  Thoughts:  If we suppose that no such $k$ exists, then every subset of the form $\{v, \ Tv,..T^{l-1}v \}$ is linearly dependent, or it is linearly dependent with $T^lv$ not lying in it span.  There are in effect then two cases to consider.  Suppose every subset of the form $\{v, \ Tv,..T^{l-1}v \}$ is linearly dependent. Then $T^{l-1}$ can be expressed as a linear combination of $v,...,T^{l-2}v$. Now, $\{v, \ Tv,..T^{l-2}v \}$ is also linearly dependent by assumption and so we can express $T^{l-2}$ as a linear combination of  $v,...,T^{l-3}v$ etc... Continuing in this way, we show that $T^{l-1}v= \lambda v$ for some $\lambda \in \mathbb{R}$. Thus $$B=\{ \mu v \ | \ \mu \ \text{is an eigenvalue of} \ T^mv \ \text{for some} \ m \}$$ This is 'at best' countably infinite, and therefore could not span $V$ even if it were 1-dimensional, wince our base field is $\mathbb{R}$, which is uncountable.  Suppose then that $B$ is linearly independent for some $k$ but that $T^kv$ does not lie in its span. I can't see how to progress this argument.  Note: This question is for second year Undergraduates, and I think there must be some more simple approach; even the first part surprises me if we need to argue using the uncountability of $\mathbb{R}$, so any alternative approaches are also appreciated.
Profile changes in Q&A are not pushed to Area 51 AirPort Time Capsule wifi is suddenly very slow: Boom network load Prove by induction. How do I prove $\sum\limits_{i=0}^n2^i=2^{n+1}-1$ with induction? The comment with the Flagged Posts that seems to be new as of today states there are posts with flags awaiting my attention but the counter shows zero and when I go into flagged posts there is nothing waiting for me. What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &amp;=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &amp;=\int_0^\infty \left(\frac{1}{xe^x} - \frac{1}{e^x-1} \right)\,dx\tag2\\ &amp;=-\int_0^1 \left(\frac{1}{1-x} + \frac{1}{\ln x} \right)\,dx\tag3\\ &amp;=\int_0^\infty \left( e^{-x} - \frac{1}{1+x^k} \right)\,\frac{dx}{x},\qquad k&gt;0\tag4\\ \end{align}$$  This is not homework problems and I know that the above integrals equal to negative of the Euler–Mascheroni constant. I got these integrals while reading this Wikipedia page: . According to Wikipedia, the Euler–Mascheroni constant is defined as the limiting difference between the harmonic series and the natural logarithm: $$\gamma=\lim_{N\to\infty} \left(\sum_{k=1}^N \frac{1}{k} - \ln N\right)$$ but I don't know why can this definition be associated to the above integrals?  I can obtain the equation $(1)$ using substitution $t=\ln \left(\frac{1}{x}\right)\rightarrow x=e^{-t} \rightarrow dx=-e^{-t}\,dt$ and I know that $$\int_0^\infty e^{-x} \ln x\,dx=\Gamma'(1)=\Gamma(1)\psi(1)=-\gamma$$ but I can't obtain the rest. Any idea? Any help would be appreciated. Thanks in advance. How do I resize/change resolution of pictures easily? Transpose/Unzip Function (inverse of zip)? Does $E = mc^2$ apply to photons? How to add custom objects to the "add object" menu? MySql autoincrement counter resetting after server restart I need to make a converter to go from RS422 full duplex to my Arduino. I have found lots of different chips which I think can do this, but am slightly confused by the terminology. The RS422 device is a laser distance sensor ().  It has RX+, RX-, TX+, TX-. So I have two lines for input and two lines for output.  I could use two RS485 line drivers to achieve the full duplex comms which I need.  There are a lot of chips to do that.  But then on my Arduino (clone: chipkit max32) would I need to use two UART channels? And then alternate which one I am sending each byte on?  The device uses 4 byte instructions. So if I wanted to send an instruction would I send Serial1(firstByte) then Serial2(secondByte), Serial1(thirdByte), Serial2(fourthByte)?   And then receive data the same way? I feel like I am missing something and can do this on one UART channel but am not sure how. says:     squinch (skwɪntʃ)      (US)      transitive verb         to squint (the eyes); squinched up her eyes in disgust.               2.         a. to pucker or screw up (the face, nose, brow, etc.); Jonathan squinched his face into a grimace.       b. to squeeze or compress; She squinched the paper up in an angry twist and stood to throw it into the stove ; Robert squinched the ketchup bottle, trying to get the last bit out.      intransitive verb         to squint, pucker, or contort   to crouch down or draw oneself together so as to seem smaller; ;  (Source: .)   to flinch         Word Origin: probably blend of squint and pinch      First Known Use: 1835      Example Sentences Including "squinch:"      Cooper saw her squinch up her eyes, trying to make out who was standing outside her kitchen window.      An unhappy smile is still a smile, insofar as the corners of the mouth are drawn back or upturned. However, the eyebrows may be furrowed, and the eyes may be squinched or teary. Also, the lips tend to be pressed together, rather than relaxed as in a happy smile.    My question is, how do native speakers of AmEng use the verb squinch in the senses supported above by CAED?  In addition, what's the difference between using either "squinch" or "squint/flinch" as in the following examples:     She squinched (up) her eyes, trying to make out who was standing outside.      She squinted her eyes, trying to make out who was standing outside.   -and-     The driver squinched as the sun hit his windshield.         -and-              -and-     He squinched as the cold water struck him.         -and- Correct usage of "see" vs. "watch" OneDrive for Windows 10 - Remove logged out account I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$ I have a Shapefile but its coordinate system is Unknown, and there is no *.prj file.   How can I identify it now? Is it fine to use due to in place of because of ? How about the other way around? Are any of these sentences ungrammatical?   He was lost because of the storm. He was lost due to the storm. He lost his way due to the storm. He  lost his way because of the storm. You have already activated X, but your Gemfile requires Y Canonical URL for a home page and trailing slashes I'm using Dataloader to Upsert a Child Account to Parent Account using an external ID.  When the Upsert is complete, the Child overwrites the parent, it's not not connecting the Child Account to the Parent Account.  ((See Answer Below))