Is there a Linear Transformation

In summary, the conversation discusses the existence of a linear transformation from R^3 to R^2 with specific coordinates. The requirements for a linear transformation are discussed, including the use of linearly independent vectors to span R^3 and the resulting transformation of a linear combination of these vectors. It is noted that the transformation is not unique and can go to any place in R^2.
  • #1
Chillguy
13
0

Homework Statement


From Hoffman and Kunze:

Is there a linear transformation T from [tex]R^3[/tex] to [tex]R^2[/tex] such that T(1,-1,1)=(1,0) and T(1,1,1)=(0,1)?

Homework Equations


[tex] T(c\alpha+\beta)=cT(\alpha)+T(\beta) [/tex]

The Attempt at a Solution


I don't really understand how to prove that there is a linear transformation with these coordinates. I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1). Then I don't really know where to go from here.
 
Physics news on Phys.org
  • #2
Chillguy said:
I think I begin by defining another arbitrary, linearly independent, transformation such as T(1,0,0)=(1,1).

Think about how this helps you along with the requirement that T is a linear transformation. For example, what does your requirement that having the transformation of another linearly independent vector in R3 give you?
 
  • #3
Orodruin said:
Think about how this helps you along with the requirement that T is a linear transformation. For example, what does your requirement that having the transformation of another linearly independent vector in R3 give you?
It would allow me to span R3 with these three vectors
 
  • #4
Chillguy said:
It would allow me to span R3 with these three vectors

And thus the transformation of any vector in R3 can be written as ...
 
  • #5
... can be written as [tex]T(c\alpha_i+\beta_i)?[/tex]
where alpha and beta are vectors in R3
 
  • #6
Well, you need to define what your ##\alpha## and ##\beta## etc are. You have three vectors with which you can express any vector in R3 as a linear combination. What will be the resulting transformation of such a linear combination? Will it fulfil the requirements you have?
 
  • #7
The resulting combination will be a vector in R^2, I think it could be any vector in R^2?
 
  • #8
No, not if you already have fixed it for three vectors. Use the linear property of T! Let us say you have the vectors ##v_1, v_2, v_3## and have fixed ##T(v_i) = u_i## for ##i = 1,2,3##. Now you take a linear combination of those ##v = \sum_i c_i v_i##. What is ##T(v)##?
 
  • #9
[tex] T(v)=T(c*v_1+c*v_2+c*v_3)[/tex]
Which I can then apply the definition of a linear transformation as defined before?
 
  • #10
Yes. So does this linear transformation fulfil the requirements?
 
  • #11
Yes! it does! Thank you so much. It makes much more sense now. I just have one other question, why do the vectors need to span all of R^3 in the first place? Is it so the transformation can go to any place in R^2?
 
  • #12
Unless you specify the transformation for any arbitrary vector in R3 you have not really found a linear transformation from R3 to R2 but only from a two-dimensional subspace. A more complete answer would be: Yes, it exists, but is not unique. (You could have selected any vector in R2 as the image of your third linearly independent vector in R3).
 

FAQ: Is there a Linear Transformation

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the basic structure of the space. In simpler terms, it is a transformation that takes one set of points and transforms them into another set of points without changing their overall shape.

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

A transformation is considered linear if it satisfies two properties: 1) it preserves addition, meaning that transforming the sum of two vectors is equal to the sum of the transformed vectors, and 2) it preserves scalar multiplication, meaning that transforming a vector multiplied by a scalar is equal to the scalar multiplied by the transformed vector. These properties can be verified by applying the transformation to a set of points and checking if they still satisfy these conditions.

3. Why is linearity important in mathematics?

Linearity is important because it allows for simpler and more efficient mathematical operations. It also helps to generalize concepts and make them applicable to a wider range of situations. Many mathematical models, such as those used in physics and economics, rely on linearity to simplify complex systems and make predictions.

4. What are some real-life examples of linear transformations?

Some real-life examples of linear transformations include scaling, rotation, reflection, and translation. These transformations are commonly used in computer graphics, engineering, and physics to manipulate and analyze data. For instance, a map projection is a linear transformation that converts points on a spherical globe to points on a flat map.

5. Can a linear transformation also be non-linear?

No, a transformation cannot be both linear and non-linear. A transformation is either linear or non-linear, there is no in-between. However, a transformation can be approximated by a linear transformation, meaning that it behaves like a linear transformation for small changes in input values. This is often done in numerical methods to simplify calculations.

Back
Top