Linear Algebra: Kernel & Range of Linear Transformation

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SUMMARY

The discussion focuses on the significance of the kernel and range (image) of a linear transformation in linear algebra. Specifically, for a linear map f: V → W, the kernel is defined as $$\ker( f)=\{0\}$$, which indicates that the map is injective. Additionally, the relationship $$V/\mbox{ker}( f) \cong \mbox{Im}( f)$$ highlights the utility of working with the image rather than the quotient space, simplifying many linear algebra problems.

PREREQUISITES
  • Understanding of linear transformations and their properties
  • Familiarity with the concepts of kernel and image in linear algebra
  • Knowledge of quotient spaces and isomorphisms
  • Basic proficiency in mathematical notation and proofs
NEXT STEPS
  • Study the properties of linear maps and their injectivity
  • Explore the concept of quotient spaces in greater detail
  • Learn about isomorphisms and their applications in linear algebra
  • Investigate additional results related to kernel and image in linear transformations
USEFUL FOR

Students of linear algebra, educators teaching linear transformations, and mathematicians interested in the foundational concepts of vector spaces and mappings.

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Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?
 
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They can be useful. Suppose you have a linear map f: V \to W. If you want to know if this linear map is injective (i.e one-to-one map) then you can take a look at the kernel: $$\ker( f)=\{0\} \Leftrightarrow \ f \ \mbox{is injective}$$

There's also the following result.
$$V/\mbox{ker}( f) \cong \mbox{Im}( f)$$
which can be very useful because it's easier to work with the image in stead of the quotientspace $V/\mbox{ker}( f)$

These are offcourse a lot of other results but these two are the first I could remember immediately.
 

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