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 Nov25 asked Proving statements related to open sets Nov25 accepted Show that this is a vector space and determine the dimension Nov25 comment Show that this is a vector space and determine the dimension @Arturo: wow, thank you! this is extremely helpful, i'll be digesting it for a little while... Nov24 comment Show that this is a vector space and determine the dimension @Arturo that'd be great, yeah it's absolutely no rush, but thank you very much. Nov24 accepted which of the following are linear mappings? Nov24 comment which of the following are linear mappings? right, that absolutely makes sense to me. i think the only thing i'm still not understanding is how to interpret (c) and (d) (without even trying to test if they are linear). i wasn't sure if the brackets around $t$ meant something in particular and i need to do some reviewing/learning of converging sequences, limits, etc... Thanks a lot for the response! Nov24 comment Show that this is a vector space and determine the dimension Let $T,U \in \mathbf{V}$, $T(a,b,c,d) + U(a,b,c,d) = (T+U)(a+a, b+b, c+c, d+d) \Rightarrow (T+U) \in \mathbf{V} \Rightarrow (T+U)$ is a linear map?? just to be clear, must the input for both $T$ and $U$ be the same? Nov24 comment Show that this is a vector space and determine the dimension @Arturo so i think it's progress that i am seeing this as the set of linear maps from $\mathbb{R}^{4}$ to $\mathbb{R}^{2}$ for which $f(v)=0$. it wouldn't be that difficult for me to accept the properties of a vector space, but i find it very difficult to begin proving that they are true. for instance to 'prove' that $T + U$ are linear whenever $T$ and $U$ are linear, i feel like i would simply be restating using symbols if i were to write something like: (continued) Nov24 revised which of the following are linear mappings? edited tags Nov24 comment Show that this is a vector space and determine the dimension @Willie thanks for the hint, i think i understand why the linear map is the set of $4 \times 2$ matrices. however i must be missing something when i try to thing of the general form of a matrix you describe since i would thing matrix $A$ could be full of zeros.. Nov24 comment Show that this is a vector space and determine the dimension @Rahul thanks for the comment, it definitely helps to clarify my conception of the given information! Nov23 asked Show that this is a vector space and determine the dimension Nov23 comment Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ In any case it was good to try to see it from another angle, although to be honest, i'm still having some trouble really visualizing this, hopefully it hits me later... and yes the matrices seem really efficient but at least for me they still look like magic ;) Nov23 accepted Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ Nov23 comment Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ @Chandru1 thanks a lot for the help i think i'm slowly starting to understand this stuff. To prove it with $n=2$ i simply broke $F_{n+2}$ down into $F_{n+1} + F_n = F_n + F_{n-1} + F_n$ which is what i obtained when substituting $n=2$. Nov23 comment Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ that is, have i understood it correctly if i were to say that to define a sequence, one creates a $2 X 2$ matrix with the upper left value equal to the next value the bottom left and upper right values equal to the current value, and the bottom right equal to the previous? Nov23 comment Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ ok so what i compute is: $$M_{n+m} = M_n*M_m = \left(\begin{array}{ccc} F_{n+1} & F_n \\\ F_n & F_{n-1} \end{array}\right)*\left(\begin{array}{ccc} F_{m+1} & F_m \\\ F_m & F_{m-1} \end{array}\right) = \left(\begin{array}{ccc} F_{n+1}F_{m+1} + F_nF_m & F_{n+1}F_m + F_nF_{m-1} \\\ F_nF_{m+1} + F_{n-1}F_m & F_{n}F_{m} + F_{n-1}F_{m-1} \end{array}\right)$$ which as you said shows the desired expression in the bottom left(and i assume equivalent) in the top right. is this method something one can usually use with sequences? i guess i just don't know how it works in general... Nov23 comment Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ i'm sorry, but i'm having trouble figuring out how to use that to simplify.... Nov23 awarded Commentator Nov23 comment Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$ am i interpreting this properly if i write: $\frac{\Phi^{n+m} - (1-\Phi)^{n+m}}{\sqrt{5}} = \frac{\Phi^{n-1} - (1-\Phi)^{n-1}}{\sqrt{5}} * \frac{\Phi^{m} - (1-\Phi)^{m}}{\sqrt{5}} + \frac{\Phi^{n} - (1-\Phi)^{n}}{\sqrt{5}} * \frac{\Phi^{1+m} - (1-\Phi)^{1+m}}{\sqrt{5}}$ $\frac{\Phi^{m} - (1-\Phi)^{m}}{\sqrt{5}}(\frac{\Phi^{n-1} - (1-\Phi)^{n-1}}{\sqrt{5}} + \frac{\Phi^{n} - (1-\Phi)^{n}}{\sqrt{5}} * F_1)$ $\frac{\Phi^{m} - (1-\Phi)^{m}}{\sqrt{5}}(\frac{\Phi^{n-1} - (1-\Phi)^{n-1}}{\sqrt{5}}+\frac{\Phi^{n} - (1-\Phi)^{n}}{\sqrt{5}})$ ? what can i do at the end?