Squaring a linear transformation

Click For Summary
SUMMARY

The discussion centers on proving that T² is a linear transformation if T is a linear transformation from R³ to R³. The key insight is that squaring a transformation refers to applying the transformation T twice, represented mathematically as T²(x) = T(T(x)). This property is essential for establishing the linearity of T², leveraging the definition of linear transformations.

PREREQUISITES
  • Understanding of linear transformations in vector spaces
  • Familiarity with the properties of functions and composition
  • Knowledge of R³ as a vector space
  • Basic proof techniques in linear algebra
NEXT STEPS
  • Study the properties of linear transformations in R³
  • Learn about function composition and its implications for linearity
  • Explore examples of linear transformations and their compositions
  • Investigate the implications of linearity in higher-dimensional vector spaces
USEFUL FOR

Students of linear algebra, mathematicians, and educators seeking to deepen their understanding of linear transformations and their properties.

starcoast
Messages
8
Reaction score
0

Homework Statement


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


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!
 
Physics news on Phys.org
starcoast said:

Homework Statement


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


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
 
Ray Vickson said:
T^2(x) = T(T(x)).

RGV

Thank you! I should have guessed that.
 

Similar threads

The correct way to write the range of a linear transformation
  • · Replies 11 ·
Replies
11
Views
2K
How to determine if a transformation is linear
  • · Replies 26 ·
Replies
26
Views
3K
Linear algebra invertible transformation of coordinates
  • · Replies 2 ·
Replies
2
Views
1K
Linear transformation: Find the necessary quantity of T
  • · Replies 3 ·
Replies
3
Views
2K
[LinAlg] Show that T:C[a,b] -> R is a linear transformation
  • · Replies 2 ·
Replies
2
Views
1K
Linear Algebra - Linearity of a transformation
  • · Replies 4 ·
Replies
4
Views
2K
Linear operator in 2x2 complex vector space
  • · Replies 18 ·
Replies
18
Views
3K
Is there a Linear Transformation
  • · Replies 11 ·
Replies
11
Views
2K
Laplace transform of cosine squared function
  • · Replies 1 ·
Replies
1
Views
1K
[Linear Algebra] Help with Linear Transformation exercises
  • · Replies 11 ·
Replies
11
Views
2K