Linear Transformation: Finding the Transformation of a Line

In summary, the conversation is about a math problem involving the transformation of a line and determining if the transformed line is parallel to the original line. The attempt at a solution involved finding the equation for the transformed line and then substituting values to see if it matched the given equation. The summary also includes a correction to the initial attempt at the solution.
  • #1
odolwa99
85
0
Having a bit of trouble with this one. Can anyone help?

Many thanks.

Homework Statement



Q. L is the line x - y + 1 = 0. f is the transformation f: (x, y) ---> (x', y') where: x' = 2x - y & y' = y. Find f(L) and investigate if f(L) is parallel to L.

Homework Equations




The Attempt at a Solution



If y = y', then x' = 2x - y => 2x = x' - y => 2x = x' - y' => x = [itex]\frac{x' - y'}{2}[/itex]

If L = x - y + 1 = 0, then f(L) = [itex]\frac{x' - y'}{2}[/itex] - y' + 1 = 0 => x' - y' - 2y' + 2 = 0 => x' - 3y' + 2 = 0

Now apply x/ y values to f(L) = x' - 3y' + 2 = 0 => 2x - y - 3y + 2 = 0 => 2x - 4y + 2 = 0 => x - 2y + 1 = 0 => 2y = x + 1 => y = [itex]\frac{x - 1}{2}[/itex]

Ans: (From textbook): y = x + 2
 
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  • #2
You went off course on the first line. If x'=2x-y' then 2x=x'+y'.
 
  • #3
Great, thank you.
 

FAQ: Linear Transformation: Finding the Transformation of a Line

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another vector space in a way that preserves the structure of the original space. In other words, it is a transformation that preserves straight lines and parallelism.

2. How do you find the transformation of a line?

To find the transformation of a line, you need to determine the transformation matrix that maps the coordinates of the original line to the coordinates of the transformed line. This can be done by using a system of linear equations and solving for the transformation matrix.

3. What are the properties of a linear transformation?

The properties of a linear transformation include preserving addition, scalar multiplication, and the zero vector. This means that the transformation of the sum of two vectors is equal to the sum of the individual transformations, the transformation of a scaled vector is equal to the scaled transformation of the original vector, and the transformation of the zero vector is equal to the zero vector.

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

A transformation is linear if it satisfies the properties of a linear transformation, which include preserving addition, scalar multiplication, and the zero vector. Additionally, a linear transformation can be represented by a transformation matrix that follows certain rules, such as having the same number of rows and columns and being able to perform matrix multiplication.

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

No, a non-linear function cannot be a linear transformation. Linear transformations are defined by their ability to preserve straight lines and parallelism, which is not possible with a non-linear function. Additionally, a linear transformation must follow certain rules and properties, which a non-linear function does not satisfy.

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