Linear transformation - adding and subtracting?

In summary, the conversation discusses a linear transformation problem where T(3 − x + 4x^2) = 1 + x − x^2 and T(2 − 3x + 2x^2) = 7 + 3x + 2x^2. The goal is to find T(7x + 2x^2) by solving a linear system of equations using the given information. The problem is solved in the end.
  • #1
Niles
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[SOLVED] Linear transformation - adding and subtracting?

Homework Statement



Suppose T : P2 -> P2 is a linear transformation satisfying T(3 − x + 4x^2) = 1 + x − x^2 and
T(2 − 3x + 2x^2) = 7 + 3x + 2x^2.
Find T(7x + 2x^2).

The Attempt at a Solution



First of all, it's linear. To find T(2x^2), I can say -x^2-2x^2 = -3x^2 = T(2x^2)? And similarily for T(7x)?
 
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  • #2
No wait, I solved it.. I can't do what I did in my original post.

I have to solve a linear system of equations to find T(7x + 2x^2) written from T(3 − x + 4x^2) and T(2 − 3x + 2x^2).

Bottom line is, I solved it.
 

Related to Linear transformation - adding and subtracting?

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space into another vector space in a way that preserves the structure of the original space. In simpler terms, it is a function that takes in a set of points and transforms them into another set of points, while maintaining their linearity.

2. How do you add two linear transformations?

To add two linear transformations, you simply add the corresponding elements of the transformation matrices. This means that the first element of the first matrix is added to the first element of the second matrix, the second element of the first matrix is added to the second element of the second matrix, and so on.

3. Can you subtract two linear transformations?

Yes, you can subtract two linear transformations by subtracting the corresponding elements of the transformation matrices. This means that the first element of the first matrix is subtracted from the first element of the second matrix, the second element of the first matrix is subtracted from the second element of the second matrix, and so on.

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

A transformation is considered linear if it satisfies two properties: 1) it preserves addition (f(u + v) = f(u) + f(v)) and 2) it preserves scalar multiplication (f(ku) = kf(u)) for all vectors u and v and scalar k. Additionally, the transformation must also map the zero vector to the zero vector (f(0) = 0).

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

A linear transformation follows the properties mentioned above, while a nonlinear transformation does not. In a nonlinear transformation, the output is not directly proportional to the input, and it may not preserve the structure of the original space. This means that lines may not remain lines and parallel lines may not remain parallel after the transformation takes place.

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