Question

Can you help me out with this question

The function 1,cos(x) , sin(x) , cos(2x), sin(2x),……. form the Fourier basis ( an infinite dimensional function space ). a)- write the 5x5 differentiation matrix for the first derivative of only those first five functions as a combination of the same five function b)- what is the kernel ( basis of null space) c)- what is image ( basis of the columns space)..
If you can please help me with part (a) i can figure out part b and c

yama_aryayee@yahoo.com

Answer 1

Is there any clarification of combination? Does the problem mean arithmetic combination of those five functions? I can see building a 5x5 matrix that way, but not sure otherwise.

Answer 2

I think it mean arithmetic . can you build the matrix and send it to me plz

Answer 3

I did n’t understand by combination , and the question is all written above , i did not miss any par

Answer 4

Again, I'm not 100% sure what the problem means by combination. The problem could mean linear combination (but I don't see how to build a 5x5 matrix using all 5 as linear combinations). So, I'm going to assume arithmetic combination, which there are 20 possible permutations (5 functions, 4 operations, 5*4 = 20 different permutations).

Well, it's quite a pain to build. I'll out line it for you. I'm going to rewrite the functions as 1,2,3,4, and 5 so I don't have to type as much. Just substitute in the numbers for their function:

[ 1 1+2-3*4/5 1+3-4*5/2 1+4-5*2/3 1+5-2*3/4
2 2+3-4*5/1 x x x
3 3+4-5*1/2 x x x
4 4+5-1*2/3 x x x
5 5+1-2*3/4 x x x ]

Where x is, just follow the pattern to fill the rest of them in. But, this is the matrix. You want the differentiation matrix. So, take the derivative of each entry. That would be your differentiation matrix.

Answer 5

Where are sines and Cosines , i become alittle confuse with can you elaborate

Answer 6

1 =1;
2 = cos(x);
3 = sin(x);
4 = cos(2x);
5 = sin(2x);

1,2,3,4, and 5 are just dummy variables so I didn't have to write as much. This is a bit ironic, actually.

Answer 7

Thanks, i will see what i can do with

Answer 8

OK. basically u are considering the (sub)space
\[ \large V=\langle1,\cos x,\sin x,\cos(2x),\sin(2x)\rangle \]
So let's consider the linear operator \(D:V\to V\) given by \(T(u)=du/dx\). therefore:
\[ \large T(\color{red}{1})=0 \]
\[ \large T(\color{red}{\cos x})=-\sin x=(-1)\cdot\sin x \]
\[ \large T(\color{red}{\sin x})=\cos x=1\cdot(\cos x) \]
\[ \large T(\color{red}{\cos(2x)})=-\sin(2x)\cdot(2)=(-2)\cdot\sin(2x) \]
\[ \large T(\color{red}{\sin(2x)})=\cos(2x)\cdot(2)=2\cdot\cos(2x) \]

Answer 9

from this we have that the matrix of this operator is
\[ \large [T]=\begin{pmatrix}
0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\
0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2\\
0 & 0 & 0 & -2 & 0
\end{pmatrix} \]

Answer 10

is this matrix is combination of the same five function ? and i think you took the derivative of each function so i can move on the finding null kernel and image of this matrix .. am i right?

Answer 11

yes

Answer 12

can you elaborate about order of pivots , because they are not in order ,,

Answer 13

it is the matrix of an operator. what do u mean by "they're not in order"?

Answer 14

and also first row is zero , however in the first derivative just our first columns should be zero

Answer 15

these are not polynomials!!!!

Answer 16

how can know the location of each pivots , because sine and cosine have opposite derivative and their derivative is interchangeable with a sign

Answer 17

u have an ordered basis.

Answer 18

listen.
i have to go.
i'll try to log-in later.

Answer 19

tnx,
form your assistance , wish you all the best

Answer 20

I THINK THE FINDAL ANSWER IS CORRECT ,, BUT I HAVE NO IADEA ABOUT ORDER OF PIVOTS