Finding Inverse of a Transformation

The summary is: In summary, the conversation discusses constructing a matrix for an arbitrary polynomial in P_2 (R) and finding its inverse using elementary row reduction. The resulting matrix S is confirmed to be the inverse of T by checking T*S = I.
  • #1
TranscendArcu
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Homework Statement



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The Attempt at a Solution


Suppose we take an arbitrary polynomial in [itex]P_2 (R)[/itex], call this [itex]a_0 + a_1 x + a_2 x^2[/itex]

[itex]T(a_0 + a_1 x + a_2 x^2) = (a_0, a_0 + a_1 + a_2, a_0 - a_1 + a_2)[/itex]

Now, I was under the impression that I could construct a matrix for T by showing what T does to each of the standard basis vectors of [itex]P_2 (R)[/itex], these being the set [itex]1,x,x^2[/itex].

Doing some mapping: [itex]T(1 + 0x + 0x^2) = (1,1,1), T(0 + 1x + 0x^2) = (0,1,-1), T(0+0x+1x^2) = (0,1,1)[/itex].

Thus my matrix for T has these outputs as its columns. Via some elementary row reduction I concluded that the matrix (with semicolons indicating the end of a row) [1 0 0;0 1/2 1/2;-1 1/2 -1/2] is the inverse of T, call it S.

Moreover, it can be checked that T*S = I, which leads me to suspect that I have the write answer. Does this look about right?
 
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  • #2
Yes it does
 

FAQ: Finding Inverse of a Transformation

1. What is the purpose of finding the inverse of a transformation?

The inverse of a transformation is used to undo the effects of the original transformation. This can be useful in solving equations, simplifying calculations, and understanding the relationship between the original and transformed data.

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

To find the inverse of a transformation, you need to use the inverse operation of the original transformation. For example, if the original transformation involved multiplying by 3, the inverse transformation would involve dividing by 3. You can also use a matrix or function to determine the inverse of a transformation.

3. Can any transformation have an inverse?

No, not all transformations have an inverse. A transformation can only have an inverse if it is a one-to-one function, meaning that each input has a unique output. If a transformation is not one-to-one, it cannot be inverted as multiple inputs can result in the same output.

4. How do you know if a transformation has an inverse?

You can determine if a transformation has an inverse by checking if it is a one-to-one function. This can be done by graphing the transformation or by using algebraic methods such as the horizontal line test. If a horizontal line intersects the graph of the transformation at more than one point, it is not one-to-one and does not have an inverse.

5. How is finding the inverse of a transformation useful in real-world applications?

Finding the inverse of a transformation is useful in many real-world applications such as engineering, physics, and data analysis. It can be used to solve equations, model relationships between variables, and simplify complex calculations. For example, in physics, finding the inverse of a velocity transformation can help determine the distance traveled by an object over a period of time.

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