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seen Sep 30 at 3:10

Nov
25
asked Proving statements related to open sets
Nov
25
accepted Show that this is a vector space and determine the dimension
Nov
25
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...
Nov
24
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.
Nov
24
accepted which of the following are linear mappings?
Nov
24
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!
Nov
24
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?
Nov
24
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)
Nov
24
revised which of the following are linear mappings?
edited tags
Nov
24
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..
Nov
24
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!
Nov
23
asked Show that this is a vector space and determine the dimension
Nov
23
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 ;)
Nov
23
accepted Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$
Nov
23
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 $n=2$.
Nov
23
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?
Nov
23
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...
Nov
23
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....
Nov
23
awarded  Commentator
Nov
23
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?