what is α? there are many possible bases for M2,2(F). assuming it is the "standard basis" Eij, then this should be [1/2 0 0 1/2] (see 2nd post below).
When gamma = {1}, I found the answer to be:
[T] = [1 0 0 1]
However, how do I calculate the answer if gamma = {2}?
Is it just [2 0 0 2]?
Thanks in advance for your help.
what is α? there are many possible bases for M2,2(F). assuming it is the "standard basis" Eij, then this should be [1/2 0 0 1/2] (see 2nd post below).
My apologies, α is actually {(1,0;1,0),(0,1;0,0),(0,1;1,0),(0,0;0,1)}, but I believe it still comes out to [2,0,0,2].
Could you confirm this is correct?
In general, I am having trouble with this type of problem when a non-standard basis is given for W. Here is another example:
T: p2(R)->R and T(f(x))=f(2)
The given bases are: B={x^2,x+1,1-x} and y(gamma)={2}
So, I need to calculate the matrix representation of T with respect to B and y.
I can easily calculate the four images:
T(x^2)= 4
T(x+1)= 3
T(1-x)= -1
So, if we had the standard basis ({1}) for W then the answer would just be:
[4 3 -1]
However, since the basis is {2}, does this mean the answer is?:
[8 6 -2]
I feel uncomfortable "just multiplying by 2", because I'm not really sure why I'm doing that.
Could you explain a little?
Thanks!
you know, using a 1-dimensional vector space is really confusing for just this reason. in the basis {2}, the number 2 has "coordinates" (1) (since 2 = (2)1).
think of it this way: when we make {2} our basis for R, we are now calling 2 our unit "length" (we're stretching the scale of our axis).
you can verify you're doing it right by checking for some easy values of p(x). let's use p(x) = x + 1. in our basis for P2, this has coordinates (0,1,0).
now we know that T(x+1) = 2+1 = 3. in the basis {2}, 3 has coordinates (3/2) (3 is 1-1/2 "twos").
so let's verify the matrix multiplication comes out right: T(0,1,0) should be (3/2).
well (8,6,-2).(0,1,0) = 6, so...oh dear. it appears that the get the "right" answer for [T], the matrix wrt the bases {x^2, x+1, 1-x}, {2},
should be [2 3/2, -1/2] (and thus the answer to your first question should be [1/2 0 0 1/2]). let's try another, more random polynomial.
let's let g(x) = 3 - 2x + 4x^2. now T(g(x)) = g(2) = 3 - 4 + 16 = 15, and in the basis {2} 15 is (15/2).
now the icky part, we have to find g(x) in terms of our basis for P2:
3 - 2x + 4x^2 = a(x^2) + b(x+1) + c(1-x) = ax^2 + (b-c)x + (b+c).
so a = 4, b-c = -2, b+c = 3.
using c = 3-b, and substituting in b-c = -2, we get: -2 = b-c = b - (3-b) = 2b - 3, so 1 = 2b, so b =1/2.
and thus c = 5/2, so 4x^2 + (1/2)(x+1) + (5/2)(1 - x) = 4x^2 + x/2 + 1/2 + 5/2 - 5x/2 = (1/2 + 5/2) - (5/2 - 1/2)x + 4x^2
= 3 - 2x + 4x^2 = g(x). so g(x) has β-coordinates (4,1/2,5/2). now to multiply by T:
(2, 3/2,-1/2).(4,1/2,5/2) = 8 + 3/4 - 5/4 = 8 - 1/2 = 15/2, the correct answer.