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Linear mapping/Linear subspaces/Kernel/Introduction/Section

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A typical property of a linear mapping is that it maps lines to lines (or to a point). More general is following statement.


Let be a field, let and denote -vector spaces and let

be a

-linear mapping. Then the following hold.
  1. For a linear subspace , the image is a linear subspace of .
  2. In particular, the image of the mapping is a linear subspace of .
  3. For a linear subspace , the preimage is a linear subspace of .
  4. In particular, is a linear subspace of .

Proof



Let denote a field, let and denote -vector spaces, and let

denote a -linear mapping. Then

is called the kernel of .

Due to the statement above, the kernel is a linear subspace of .


For an -matrix , the kernel of the linear mapping

given by is just the solution space of the homogeneous linear system

The following criterion for injectivity is important.


Let denote a field, let and denote -vector spaces, and let

denote a -linear mapping. Then is injective if and only if

holds.

If the mapping is injective, then there can exist, apart from , no other vector with . Hence, .
So suppose that , and let be given with . Then, due to linearity,

Therefore, , and so .