Thread: Operator

Need some help with this... doesn't seem to be a decent example in my book. Appreciate any help!

Originally Posted by gralla55

Need some help with this... doesn't seem to be a decent example in my book. Appreciate any help!

1) Apply the transformation T to each and every element of E and write the result as a linear combination of E itself;

2) Take the transpose , if you apply transformations from the left and vectors from the right), or directly (otherwise) the coefficients matrix of the above: this is .

For example:


...

So the first two columns of are (or the first two rows if you write the map on the right of the vector).

Tonio

Thanks for your reply! But how do you get:

T(x) = -2 + 2x ? Shouldn't that just equal 2?

Thanks again.

Originally Posted by gralla55

Thanks for your reply! But how do you get:

T(x) = -2 + 2x ? Shouldn't that just equal 2?

Thanks again.

Yes, that must have been a typo. Since T(f)= f(x)+ f(2- x), T(x)= x+ (2- x)= 2.

In that case 2 is the answer for all four. And a linear combination would just be every element of E times 1/2 times each column vector? And won't the transpose of this matrix is the same as the matrix itself?

Originally Posted by gralla55

In that case 2 is the answer for all four.


Why? ...

Tonio


And a linear combination would just be every element of E times 1/2 times each column vector? And won't the transpose of this matrix is the same as the matrix itself?

.

Don't you just substitute the "x" for "x^2" ? I don't see why the ^2 suddenly goes outside the parantheses. But of course, I might be wrong here...

What confuses me is the notation T(f(x))... In case you're right:

T(1) = 1 + (2-1) = 2
T(x) = x + (2x-2) = 2
T(x^2) = 4 - 4x + 2x^2
T(x^3) = 8 - 12x + 8x^2 - 2x^3

How to proceed?

Originally Posted by gralla55

What confuses me is the notation T(f(x))... In case you're right:

T(1) = 1 + (2-1) = 2
T(x) = x + (2x-2) = 2
T(x^2) = 4 - 4x + 2x^2
T(x^3) = 8 - 12x + 8x^2 - 2x^3

How to proceed?

Write each outcome as a linear combination of the given basis and thus the coefficients in each case become the wanted matrix's columns...

Tonio

So:

2 2 4 8
0 0 -4 -12
0 0 2 8
0 0 0 -2

and then I take the transpose:

2 0 0 0
2 0 0 0
4 -4 2 0
8 -12 8 -2

And that's it?

Nevermind, I figured it out! And I also did a mistake, the correct answer is:

2 2 4 8
0 0 -4 -12
0 0 2 6
0 0 0 0

Now, how can I find a solution to T(f) = (x-1)^2 ?