Linear Algebra question regarding linear operators and matrix rep. relative to basis


by gothloli
Tags: algebra, basis, linear, matrix, operators, relative
gothloli
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#1
Dec12-12, 07:33 PM
P: 30
1. The problem statement, all variables and given/known data
Let w = a + bi be a complex number and let T : C -> C be defined by T(z) = w z.
Considering C as a vector space over R, find the matrix B representing T relative to
the basis {1, i} of C.


2. Relevant equations



3. The attempt at a solution
I think you use eigenvalues and eigenvectors, if T is diagonalizable, but my main problem is finding standard basis of T since z is not defined, hence can you guide me there thanks, please help I have an exam tomorrow!.
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Mark44
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#2
Dec12-12, 08:07 PM
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Quote Quote by gothloli View Post
1. The problem statement, all variables and given/known data
Let w = a + bi be a complex number and let T : C -> C be defined by T(z) = w z.
Considering C as a vector space over R, find the matrix B representing T relative to
the basis {1, i} of C.


2. Relevant equations



3. The attempt at a solution
I think you use eigenvalues and eigenvectors, if T is diagonalizable
I don't see that eigenvalues or eigenvectors enter into this at all.
Quote Quote by gothloli View Post
, but my main problem is finding standard basis of T
"standard basis of T" makes no sense to me. A basis is associated with a vector space, not a transformation.

You are given a basis for C; namely {1, i}. What is T(1)? What is T(i)?
Quote Quote by gothloli View Post
since z is not defined, hence can you guide me there thanks, please help I have an exam tomorrow!.
gothloli
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Dec12-12, 08:25 PM
P: 30
sorry I meant to say standard matrix of T not basis. Then can you tell me how to solve the question please, I have an exam tomorrow, I'm so confused, I just need help.

Dick
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#4
Dec12-12, 09:15 PM
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Linear Algebra question regarding linear operators and matrix rep. relative to basis


Quote Quote by gothloli View Post
sorry I meant to say standard matrix of T not basis. Then can you tell me how to solve the question please, I have an exam tomorrow, I'm so confused, I just need help.
w=a+bi. As Mark44 suggested, if you find T(1) and T(i) then those will be the column vectors of the 2x2 matrix for T in the basis {1,i}. What are they? Express them in terms of the basis.
gothloli
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Dec12-12, 10:41 PM
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Quote Quote by Dick View Post
w=a+bi. As Mark44 suggested, if you find T(1) and T(i) then those will be the column vectors of the 2x2 matrix for T in the basis {1,i}. What are they? Express them in terms of the basis.
okay so I get (a -b)
(b a)

thanks for the help, you made it clear for me.
gothloli
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Dec12-12, 10:42 PM
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Quote Quote by Dick View Post
w=a+bi. As Mark44 suggested, if you find T(1) and T(i) then those will be the column vectors of the 2x2 matrix for T in the basis {1,i}. What are they? Express them in terms of the basis.
I get the matrix (a -b)
(b a)
I don't have time to find the matrix input on this thing.

Thanks for the help, you made it clear
Dick
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#7
Dec12-12, 10:50 PM
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Quote Quote by gothloli View Post
I get the matrix (a -b)
(b a)
I don't have time to find the matrix input on this thing.

Thanks for the help, you made it clear
I'm clear you've got it. That's what's important. Don't worry about the notation. I fudge it a lot myself. I'd express that as [[a,-b],[b,a]] and just hope people get it.


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