There is a linear transformation from P1 to P1

In summary, there is a linear transformation T from P1 to P1, where P1 is the set of all polynomials of degree at least 1, given by T(1 + 2x) = 2 + 4x and T(4 + 7x) = -2 + 2x. To find T(-3 - 5x), we first need to represent [-3, -5] in terms of the basis B1 = [1, 2] and B2 = [2, 4]. However, the attempt at a solution provided is incorrect. The correct answer is 4 + 2x.
  • #1
Axoren
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0

Homework Statement


There is a linear transformation T from P1 to P1 where P1 is the set of all polynomials of degree at least 1.
T(1 + 2x) = 2 + 4x and T(4 + 7x) = -2 + 2x

Find T(-3 - 5x).

Homework Equations


T(1 + 2x) = 2 + 4x
T(4 + 7x) = -2 + 2x

The Attempt at a Solution


Basis B1 = [1, 2] [4, 7]
Basis B2 = [2, 4] [-2, 2]

[-3, -5] in terms of Basis 1 = [1, -1], bring that into Basis 1 gives you 1*[2, 4x] + -1*[-2, 2x] = [0, 6x]

The answer is not 6x.
 
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  • #2
Why do you think 1*[2, 4x] + -1*[-2, 2x] = [0, 6x]? That doesn't look at all right.
 
  • #3
Dick said:
Why do you think 1*[2, 4x] + -1*[-2, 2x] = [0, 6x]? That doesn't look at all right.

Haven't slept in 3 days, the other day, I was trying to calculate the cross product of two vectors in R5. What's worse, is I ended up with an answer.


Close this, the answer's 4 + 2x.
 

FAQ: There is a linear transformation from P1 to P1

What is a linear transformation from P1 to P1?

A linear transformation from P1 to P1 is a mathematical function that maps one polynomial of degree 1 to another polynomial of degree 1. It is also known as a linear map or a linear operator.

How is a linear transformation from P1 to P1 represented?

A linear transformation from P1 to P1 can be represented by a matrix. The elements of the matrix correspond to the coefficients of the polynomial. For example, the linear transformation T(x) = 2x + 1 can be represented by the matrix [2 1].

What is the significance of P1 in a linear transformation from P1 to P1?

P1 refers to the vector space of polynomials of degree 1. This vector space has a basis of 1 and x, and any polynomial of degree 1 can be represented as a linear combination of these basis vectors. A linear transformation from P1 to P1 preserves this structure and maps polynomials of degree 1 to other polynomials of degree 1.

What are some examples of linear transformations from P1 to P1?

Some examples of linear transformations from P1 to P1 are multiplying a polynomial by a constant, adding a constant to a polynomial, and taking the derivative of a polynomial. For instance, the linear transformation T(x) = 3x maps the polynomial 2x + 1 to the polynomial 6x + 3.

How is a linear transformation from P1 to P1 applied in real life?

A linear transformation from P1 to P1 can be used in various fields such as engineering, physics, and computer graphics. For example, in computer graphics, linear transformations are used to rotate, scale, and translate objects on a screen. In physics, linear transformations are used to model physical systems and analyze their behavior.

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