Can a Transformation Matrix be one-to-one and not onto?

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SUMMARY

A transformation matrix can be one-to-one if its columns are linearly independent, which is confirmed by the presence of a pivot in each column. However, it can still be one-to-one without being onto, particularly when mapping an Rn vector space onto an Rm space where m < n. This means that while the transformation is injective, it does not cover the entire codomain.

PREREQUISITES
  • Understanding of linear independence
  • Familiarity with pivot positions in matrices
  • Knowledge of vector spaces Rn and Rm
  • Basic concepts of linear transformations
NEXT STEPS
  • Study the properties of linear transformations in depth
  • Learn about the Rank-Nullity Theorem
  • Explore examples of one-to-one transformations that are not onto
  • Investigate the implications of dimension in linear mappings
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Students of linear algebra, mathematicians, and anyone studying the properties of transformation matrices and their applications in vector spaces.

suchara
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Title says all..
A transformation matrix is one to one if its columns are linearly independent, meaning it has a pivot in each column
but what if it doesn't have a pivot in each row(i.e. not onto)? is it still one-to one?
 
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actually nvm,... I just realized it will map an Rn vector onto Rm where m < n, but its still one-to-one
 

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