Is L(x,y) a Linear Transformation?

In summary, the conversation discusses determining if a given function is a linear transformation. The first step is to show that L(u+v) = L(u) + L(v), and the conversation goes on to discuss how to find L(u+v) and the confusion about the domain being R^2 and the range being R^3. The final conclusion is that L(x+x',y+y') would be (x+x'+1,y+y',x+x'+y+y').
  • #1
aznkid310
109
1
[SOLVED] Linear Transformation

Homework Statement


Determine if this is a linear transformation:

L(x,y) = (x+1, y, x+y)


Homework Equations



This is just one, but I have no clue as to how to even begin. I've been to lecture and read the book over and over again, but i was not given any relevant examples. Could someone please walk me through this? I know that to show it is a linear transformation, i must show that L(u+v) = L(u) + L(v), but i can't seem to find L(u+v)


The Attempt at a Solution



u =
[x
y]

v =
[x'
y']

L(u) + L(v) =
x + x' + 2
y + y'
x + y + x' +y'

I'm not even sure that is correct, but if it is, how does one find L(u+v)? Additionally, the fact that it is a transformation from R^2 => R^3 is throwing me off
 
Last edited:
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  • #2
The domain is R^2, they mean that (x,y)+(x',y') should be defined by (x+x',y+y'). What's L of that?
 
  • #3
would L(x+x', y+y') = x+x'+2 ?
y+y'
x+1+y

It's probably extremely obvious, but i still don't understand.
 
  • #4
L(x+x',y+y') would be (x+x'+1,y+y',x+x'+y+y'). Look at the definition. Substitute x+x' -> x and y+y' ->y. Notice that's different from what you found for L(u)+L(v).
 
  • #5
ah that makes sense! I think i understand now.Thx for the help!
 

FAQ: Is L(x,y) a Linear Transformation?

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another vector space while preserving the basic structure of the original space. This means that the transformation must preserve addition and scalar multiplication, and the zero vector must map to the zero vector.

How is a linear transformation represented?

A linear transformation can be represented by a matrix. The matrix will have the same number of rows and columns as the dimensions of the vector spaces involved. Each column of the matrix will correspond to a basis vector of the input space, and the resulting vector after transformation is found by multiplying the matrix by the original vector.

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

A linear transformation follows the rules of linearity, meaning that the output is directly proportional to the input. This results in a straight line when graphed. A nonlinear transformation does not follow these rules, and the output is not directly proportional to the input. This results in a curved or non-linear graph.

What are some real-life applications of linear transformations?

Linear transformations are used in a variety of fields, including engineering, physics, economics, and computer graphics. They can be used to model systems and predict outcomes, as well as to manipulate images and create 3D graphics in computer programs.

What is the importance of linear transformations?

Linear transformations are important in mathematics and science because they allow us to represent and understand complex systems in a more simplified way. They also have many practical applications in fields such as engineering and economics, making them an essential concept to understand for many professionals.

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