Determine if a transformation is linear

In summary: The orthogonal projection is linear because "the transformation is defined by the equations" and the equation for the orthogonal projection is "T(x, y)=0".
  • #1
aero_zeppelin
86
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Well, I've done some research on the further, more advanced chapters in my course and learned that these two conditions must be met:

T(u + v) = T(u) + T(v)
T(ku) = kT(u)


I was wondering if there's another way to figure if the transformation is linear (since this question is being asked in one of the "basic" chapters previous to the one mentioning the formulas)

Thanks!
 
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  • #2
btw... What would be a good definition for LINEAR TRANSFORMATION?
 
  • #3
search linear map on wikipedia. That gives pretty much the same definition as you have, but they say it a bit more rigorously.
 
  • #4
aero_zeppelin said:
Well, I've done some research on the further, more advanced chapters in my course and learned that these two conditions must be met:

T(u + v) = T(u) + T(v)
T(ku) = kT(u)


I was wondering if there's another way to figure if the transformation is linear (since this question is being asked in one of the "basic" chapters previous to the one mentioning the formulas)

Thanks!
The point of what aerozeppelin and Bruce W are saying is that "T(u+v)= T(u)+ T(v)" and "T(ku)= kT(u)" are usually taken as the definition of "linear transformation". And you certainly can't ask whether something is a linear transformation before you have defined it! If your text is asking to determine whether a function is a linear transformation before those formulas are given, what definition is given?
 
  • #5
The first time you get the LINEAR TRANSFORMATION idea in my text, they say:

" In the special case where the equations in 1 are linear, the transformation
T: Rn --> Rm defined by those equations is called a linear transformation (or a linear operator if m = n ). "And then, the question asked is:
" Show that the orthogonal projections on the coordinate axes are linear operators, and find their standard matrices"

How can this be shown?
 

1. What is a linear transformation?

A linear transformation is a mathematical function that maps every point in one coordinate system to a corresponding point in another coordinate system in a straight line. It can be represented by a matrix multiplication and preserves the basic operations of vector addition and scalar multiplication.

2. How do you determine if a transformation is linear?

To determine if a transformation is linear, you can use the following two properties:
1. The transformation must preserve vector addition, meaning that T(a + b) = T(a) + T(b).
2. The transformation must preserve scalar multiplication, meaning that T(ka) = kT(a) where k is a constant.

3. What is the difference between a linear and non-linear transformation?

A linear transformation is one that preserves the basic properties of vector addition and scalar multiplication, while a non-linear transformation does not. Non-linear transformations can result in curved or distorted mappings between coordinate systems.

4. Can a transformation be both linear and non-linear?

No, a transformation can only be either linear or non-linear. It cannot possess properties of both at the same time.

5. What are some common examples of linear transformations?

Some common examples of linear transformations include rotation, scaling, shearing, reflection, and translation. These transformations can be represented by matrices and are often used in computer graphics, physics, and engineering applications.

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