Squaring a linear transformation

In summary, when proving that T^2 is a linear transformation if T is linear, it is helpful to remember that T^2(x) = T(T(x)). This property simplifies the proof.
  • #1
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Homework Statement


Prove that T[itex]^{2}[/itex] is a linear transformation if T is linear (from R[itex]^{3}[/itex] to R[itex]^{3}[/itex].


So I understand when a transformation is considered linear, but I don't understand what squaring a transformation does. I don't think it means squaring the result of the transformation but I'm not sure how else to think of it. Or maybe I'm forgetting some convenient property that would make this proof short and sweet. Any help is very much appreciated!
 
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  • #2
starcoast said:

Homework Statement


Prove that T[itex]^{2}[/itex] is a linear transformation if T is linear (from R[itex]^{3}[/itex] to R[itex]^{3}[/itex].


So I understand when a transformation is considered linear, but I don't understand what squaring a transformation does. I don't think it means squaring the result of the transformation but I'm not sure how else to think of it. Or maybe I'm forgetting some convenient property that would make this proof short and sweet. Any help is very much appreciated!

T^2(x) = T(T(x)).

RGV
 
  • #3
Ray Vickson said:
T^2(x) = T(T(x)).

RGV

Thank you! I should have guessed that.
 

1. What is a linear transformation?

A linear transformation is a mathematical operation that maps a vector or point in one space to another space in a linear fashion. In simple terms, it is a function that transforms a set of coordinates or values into a new set of coordinates or values.

2. What does it mean to square a linear transformation?

Squaring a linear transformation means performing the transformation twice in succession. This results in a new transformation that is equivalent to applying the original transformation twice. In other words, it is a composite transformation.

3. How is squaring a linear transformation different from composing two linear transformations?

Squaring a linear transformation is a specific case of composing two linear transformations. When squaring a linear transformation, the same transformation is applied twice. Composing two linear transformations, on the other hand, can involve different transformations being applied in succession.

4. What is the significance of squaring a linear transformation?

Squaring a linear transformation can have various applications in mathematics and science. It can be used to simplify complex transformations, solve certain differential equations, and analyze the behavior of dynamical systems.

5. Are there any limitations to squaring a linear transformation?

Yes, there are limitations to squaring a linear transformation. Not all linear transformations can be squared, and even if they can, the resulting transformation may not always be useful or meaningful. Additionally, the order in which the transformations are squared can affect the final result.

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