Linear Operator/transformation Help

  • Thread starter Spraypaint
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In summary, to compute L(1,2,3)^T, we need to express (1,2,3) as a linear combination of the given vectors, use the definition of L as a linear transformation to compute L(1,2,3)^T, and then use the values we found in the linear combination to get the final answer.
  • #1
Spraypaint
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Homework Statement


Let L: R^3 -> R^3 be a linear operator. Given that L(1,1,-1)^T = (2,3,4)^T, L(1,-1,1)^T = (1,5,1)^T, and L(-1,1,1)^T = (2,0,0)^T, Compute L(1,2,3)^T. Justify your answer.


Homework Equations


None.


The Attempt at a Solution


I honestly do not really have a clue. I have an understanding on linear transformations and how they must satisfy the addition and scalar multiplication, but I don't know how to go about doing this question.
 
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  • #2
If you understand that L is a linear transformation, then you just have to figure out a way to express (1,2,3) in the form a*(1,1,-1)+b*(1,-1,1)+c*(-1,1,1), right?
 
  • #3
I assume "^T" mean transpose. You need to write (1,2,3) as a linear combination of the other vectors you were given. Then use the fact that L is linear to compute L(1,2,3).
 
  • #4
So my understanding of the question is to find how the linear transformation in this problem is defined, and use that to solve for L(1,2,3). Is this wrong?
 
  • #5
You are given how the linear transformation is defined, i.e. the three values. But yes, you will use those definitions along with the fact that L is linear to solve for L(1,2,3).
 
  • #6
Dick said:
If you understand that L is a linear transformation, then you just have to figure out a way to express (1,2,3) in the form a*(1,1,-1)+b*(1,-1,1)+c*(-1,1,1), right?

So going on with this idea, you would get three equations ( a + b - c, a - b + c, -a +b +c). Do you set this equal to 1,2,3? Or am I completely off?
 
  • #7
Yes, you set it equal to (1,2,3) and figure out a, b and c. Then apply L to the sum.
 
  • #8
[edit]

Sorry, I think I understand now. I should do a(2,3,4) + b(1,5,1) + c(-1,1,1), and use the a,b,c values I got from the other part to solve for L(1,2,3), right?
 
  • #9
No, you need those at the end in order to get a final result for L(1,2,3).
 
  • #10
Noooo. You'll need those to get the answer. If v=a*v1+b*v2+c*v3 then L(v)=a*L(v1)+b*L(v2)+c*L(v3). You'll need L(v1), L(v2) and L(v3) to get the final answer.
 
  • #11
Thank you guys, it makes sense, you surely made my day. =)
 

What is a linear operator/transformation?

A linear operator/transformation is a mathematical function that maps one vector space to another vector space while preserving the basic operations of vector addition and scalar multiplication. In other words, it is a function that takes in a vector and produces another vector that is in the same vector space.

What is the difference between a linear operator and a linear transformation?

In most cases, the terms "linear operator" and "linear transformation" are used interchangeably. However, some sources make a distinction by defining a linear operator as a function between two vector spaces, while a linear transformation is a function from a vector space to itself.

What are some common examples of linear operators/transformations?

Some common examples include rotation, reflection, scaling, and translation in 2D or 3D space. Other examples include differentiation and integration in calculus, and matrix operations such as matrix multiplication and determinant calculation.

How can I determine if a function is a linear operator/transformation?

To determine if a function is a linear operator/transformation, you can check if it satisfies the two properties of linearity: additivity and homogeneity. Additivity means that the function satisfies f(u + v) = f(u) + f(v) for any two vectors u and v, while homogeneity means that the function satisfies f(αu) = αf(u) for any vector u and scalar α.

What are the applications of linear operators/transformations in science?

Linear operators/transformations have many applications in science, including in physics, engineering, and computer science. They are used to model and analyze physical systems, such as in quantum mechanics and fluid dynamics. In engineering, they are used for signal processing, control systems, and image processing. In computer science, they are used for data compression, computer graphics, and machine learning algorithms.

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