Find Matrix B for Complex Number T(z)

In summary, to find the matrix B representing the transformation T(z) = w · z relative to the basis {1, i} of C, we can simply evaluate T(1) and T(i) to obtain the column vectors [a, b] and [-b, a] respectively. These vectors then form the 2x2 matrix B = [[a, -b], [b, a]].
  • #1
gothloli
39
0

Homework Statement


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.


Homework Equations





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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  • #2


gothloli said:

Homework Statement


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.


Homework Equations





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.
gothloli said:
, 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)?
gothloli said:
since z is not defined, hence can you guide me there thanks, please help I have an exam tomorrow!.
 
  • #3


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.
 
  • #4


gothloli said:
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.
 
  • #5


Dick said:
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.
 
  • #6


Dick said:
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
 
  • #7


gothloli said:
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.
 

1. What is a complex number?

A complex number is a mathematical concept that combines a real number and an imaginary number. It can be expressed in the form a + bi, where a represents the real part and bi represents the imaginary part (with i being the imaginary unit).

2. What does the matrix B represent in "Find Matrix B for Complex Number T(z)"?

The matrix B represents the transformation of a complex number into another complex number using the function T(z). This transformation can be visualized as a mapping of points on a complex plane.

3. How is the matrix B calculated for a given complex number?

The matrix B for a complex number z can be calculated by taking the real and imaginary parts of z and arranging them in a 2x2 matrix as follows:
B = [ Re(z) -Im(z)
Im(z) Re(z) ]

4. Can the matrix B be used to perform operations on complex numbers?

Yes, the matrix B can be used to perform operations such as addition, subtraction, multiplication, and division on complex numbers. These operations can be carried out by multiplying the matrix B with another matrix representing the complex number being operated on.

5. How is the matrix B helpful in studying complex numbers?

The matrix B provides a visual representation of the transformation that occurs when a complex number is operated on by a function. This can help in understanding the behavior of complex numbers and their operations. Additionally, the matrix B can be used to simplify complex calculations involving multiple complex numbers.

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