Linear Transformations and their matrices

In summary, the problem is to show that T(x + yi) = x - yi is a linear transformation and to find the matrix of the transformation using the basis (1+i, 1-i). To do this, one must apply the linear transformation to each basis vector and write the result in terms of the basis. The coefficients of the linear combination form the columns of the matrix. The definition of a linear transformation must also be satisfied, where u and v are complex numbers and a is a real number.
  • #1
succubus
33
0
The problem is

T(x + yi) = x - yi

Show that this is a linear transformation and find the matrix of the transformation using the following basis

(1+i, 1-i)

ARGH


I am having trouble with the complex numbers for some reason!


To show that it is linear I have to show

T(x + yi + a + bi) = x + (-yi) + a + (-bi) = x + a + (- i(y +b)) = T(x + yi + a +bi) = x + (-yi) + a + (-bi)

T(k(x + yi)) = k(x + (-yi)) = kx + k(-yi) = T(k(x + (-yi)) = kT(x + (-yi))

Is this correct?

And as for finding the matrix. OY!

I know B = [ [T(1+i)] [T(1-i)] ]

So I know how to do it in theory, kind of I guess. But I just don't know how to start :/
 
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  • #2
Start with the definition for a linear transformation. If T is a linear transformation from C to C, what conditions have to be satisfied?
 
  • #3
Well the definition of a transformation is it's closed under addition and scalar multiplication, but I don't see how that helps. Is this regarding finding the matrix of the transformation?
 
  • #4
succubus said:
The problem is

T(x + yi) = x - yi

Show that this is a linear transformation and find the matrix of the transformation using the following basis

(1+i, 1-i)

ARGH


I am having trouble with the complex numbers for some reason!


To show that it is linear I have to show

T(x + yi + a + bi) = x + (-yi) + a + (-bi) = x + a + (- i(y +b)) = T(x + yi + a +bi) = x + (-yi) + a + (-bi)

T(k(x + yi)) = k(x + (-yi)) = kx + k(-yi) = T(k(x + (-yi)) = kT(x + (-yi))

Is this correct?

And as for finding the matrix. OY!

I know B = [ [T(1+i)] [T(1-i)] ]

So I know how to do it in theory, kind of I guess. But I just don't know how to start :/
There is a standard method for finding the matrix representation of a linear transformation in a given (ordered) basis:
Apply the linear transformation to each basis vector in turn. Write the result in terms of the basis. The coefficients are form the columns of the matrix. That's exactly what your "B= [[T(1+i)] [T(1-i)]]" means.

For example, here your given basis is 1+ i, 1- i. T(1+ i)= 1- i= 0(1+ i)+ 1(1- i). The first column of the matrix is
[tex]\begin{bmatrix}0 \\ 1\end{bmatrix}[/tex].

What is the second column?
 
  • #5
succubus said:
Well the definition of a transformation is it's closed under addition and scalar multiplication, but I don't see how that helps. Is this regarding finding the matrix of the transformation?

The only way you're going to be able to do the first part of this problem is by using the definition of a linear transformation. What you gave is the definition of a subspace of a vector space. For a linear transformation, you have to show that these conditions are satisfied:
T(u + v) = T(u) + T(v)
T(au) = aT(u)

where for your problem, u and v are complex numbers, and a is a real number.
 
  • #6
To HallsofIvy,

So you're saying we take each basis, perform the linear transformation on the basis, and then we find a linear combination of the original basis that satisfies the transformation and that gives me my column?

So the second column would be something like this

T(1-i) = (1 + i) = c1(1+i) + c2(1-i) = 1(1+i) + 0(1-i). so the second column would be

1
0
?
 

What is a linear transformation?

A linear transformation is a mathematical function that maps vectors from one vector space to another, while preserving the basic algebraic structure of the original space. In simpler terms, it is a transformation that preserves lines and the origin.

What is a matrix representation of a linear transformation?

A matrix representation of a linear transformation is a way of expressing the transformation as a matrix. This is done by assigning each vector in the original space to a column vector in the matrix, and the transformed vector is then found by multiplying the matrix by the original vector.

What are the properties of a linear transformation?

A linear transformation has three main properties: it preserves addition, it preserves scalar multiplication, and it preserves the zero vector. This means that the transformation of the sum of two vectors is equal to the sum of the transformed vectors, the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformed vector, and the transformation of the zero vector is equal to the zero vector.

What is a basis and why is it important in linear transformations?

A basis is a set of linearly independent vectors that span a vector space. It is important in linear transformations because it provides a way to represent all vectors in the space using a smaller set of basis vectors. This simplifies calculations and allows for a better understanding of the transformation.

How do you determine if a transformation is linear?

A transformation is considered linear if it satisfies the properties of linearity: preserving addition, scalar multiplication, and the zero vector. To determine if a transformation is linear, these properties can be checked by plugging in values and seeing if they hold true. Alternatively, the transformation can be represented as a matrix and checked for linearity using matrix operations.

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