Linear tranformations equality

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SUMMARY

The discussion centers on the equality of linear transformations based on their kernel and image properties. It concludes that two linear transformations, g and f, with kernels equal to <0,0,0> are not necessarily equal, as multiple transformations can share the same kernel but have different images. Conversely, if the images of both transformations are equal to <0,0,0>, then both transformations must be the zero linear map, confirming their equality. This analysis is crucial for understanding the implications of kernel and image in linear algebra.

PREREQUISITES
  • Understanding of linear transformations
  • Knowledge of kernel and image concepts in linear algebra
  • Familiarity with homogeneous systems of equations
  • Basic proficiency in vector spaces and dimensions
NEXT STEPS
  • Study the properties of linear transformations in detail
  • Learn about the Rank-Nullity Theorem
  • Explore examples of linear transformations with different kernels and images
  • Investigate the implications of zero linear maps in vector spaces
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Students of linear algebra, educators teaching vector space theory, and mathematicians analyzing linear transformations and their properties.

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Homework Statement



We are given a homogeneous system of 5 equations and 3 variables. We are asked to find the solutions (which i found to be unique and <0,0,0>) and then we are asked(along loads of other stuff:P):

a)If any two linear transformations g and f have kerf=kerg=<0,0,0> then they are equal.
b)If any two linear transformations g and f have imf=img=<0,0,0> then they are equal.

Homework Equations





The Attempt at a Solution



a) is wrong because there are infinite linear transformations with ker=<0,0,0> but different images. for (positive but not absolutely sure i am correct because the dimension of the images would be equal)
i think b) is correct but i am not sure. I would say that if imf=img=<0,0,0> then they are both the zero linear map so they are equal. Thanks again in advance
 
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