Linear Transformations (or lack thereof)

In summary, the complex linear map is a map of the form z \mapsto az for some a \in \mathbb{C}, ie. multiplication by a constant. This map is not complex linear if V is regarded as a vector space over the complex numbers. However, it is real linear (just (x,y) \mapsto (x, -y)).
  • #1
nateHI
146
4

Homework Statement


Let V be the set of complex numbers regarded as a vector space over the real numbers R. Find a linear transformation T: V → V which is not complex linear (i.e. not a linear transformation if V is regarded as a vector space over the complex numbers).


Homework Equations



V=R(i)

The Attempt at a Solution


I have a solution. What I need is a better explanation of what complex linear means. Or perhaps a rewording of the problem statement so I might have a better understanding of why my solution is a solution.
 
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  • #2
hi nateHI! :smile:
nateHI said:
What I need is a better explanation of what complex linear means. Or perhaps a rewording of the problem statement so I might have a better understanding of why my solution is a solution.

f is linear if f(λa + µb) = λf(a) + µf(b) for all scalars λ and µ

so what happens eg if λ = i ? :wink:
 
  • #3
"... regarded as a vector space over the real numbers R"

What does that mean?
 
  • #4
I think they mean z = u +iv with (u,v) in RxR.
 
  • #5
nateHI said:

Homework Statement


Let V be the set of complex numbers regarded as a vector space over the real numbers R. Find a linear transformation T: V → V which is not complex linear (i.e. not a linear transformation if V is regarded as a vector space over the complex numbers).


Homework Equations



V=R(i)

The Attempt at a Solution


I have a solution. What I need is a better explanation of what complex linear means. Or perhaps a rewording of the problem statement so I might have a better understanding of why my solution is a solution.

The complex numbers are a 1-dimensional vector space over themselves, so a complex-linear map [itex]\mathbb{C} \to \mathbb{C}[/itex] is a map of the form [itex]z \mapsto az[/itex] for some [itex]a \in \mathbb{C}[/itex], ie. multiplication by a constant.

The map which takes [itex]z = x + iy[/itex] to its complex conjugate [itex]z^{*} = x - iy[/itex] is not of that form (there is no [itex]a \in \mathbb{C}[/itex] such that [itex]z^{*} = az[/itex] for all [itex]z \in \mathbb{C}[/itex]), and so is not complex linear. But it is real linear (it's just [itex](x,y) \mapsto (x, -y)[/itex]).

Therefore the map which takes [itex]z[/itex] to its real part [itex]\frac12 (z + z^{*})[/itex] is not complex linear either.
 
  • #6
pasmith said:
The map which takes [itex]z = x + iy[/itex] to its complex conjugate [itex]z^{*} = x - iy[/itex] is not of that form (there is no [itex]a \in \mathbb{C}[/itex] such that [itex]z^{*} = az[/itex] for all [itex]z \in \mathbb{C}[/itex]), and so is not complex linear. But it is real linear (it's just [itex](x,y) \mapsto (x, -y)[/itex]).

No not true, take z = i then z* = -i, take a =-1 and you got it.
 
  • #7
dirk_mec1 said:
No not true, take z = i then z* = -i, take a =-1 and you got it.

Your [itex]a[/itex] must work FOR ALL [itex]z \in \mathbb{C}[/itex]! Finding an [itex]a[/itex] which works for one [itex]z[/itex] is irrelevant!
 
  • #8
You're right my bad.
 
  • #9
OK, I get it now. Thanks pasmith and everyone else.
 

Related to Linear Transformations (or lack thereof)

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving certain properties such as linearity and proportionality.

2. How is a linear transformation different from a non-linear transformation?

A linear transformation follows the rules of linearity, meaning that the transformation is consistent and proportional regardless of the input. A non-linear transformation does not follow these rules and may produce varying outputs for the same input.

3. What are some common examples of linear transformations?

Some common examples of linear transformations include scaling, rotation, and reflection in geometry, as well as matrix operations such as addition, multiplication, and inversion.

4. How do linear transformations apply to real-world problems?

Linear transformations are used to model and solve a variety of real-world problems, including in economics, physics, and computer graphics. For example, linear transformations can be used to analyze changes in supply and demand, the motion of objects, and the manipulation of images in computer programs.

5. Can all transformations be represented as linear transformations?

No, not all transformations can be represented as linear transformations. In order for a transformation to be considered linear, it must follow specific rules of linearity. If a transformation does not follow these rules, it cannot be represented as a linear transformation.

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