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Linear algebra: Prove the statement

  1. Feb 22, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that [itex]\dim L(\mathbb F)+\dim Ker L=\dim(\mathbb F+Ker L)[/itex] for every subspace [itex]\mathbb{F}[/itex] and every linear transformation [itex]L[/itex] of a vector space [itex]V[/itex] of a finite dimension.

    2. Relevant equations
    -Fundamental subspaces
    -Vector spaces

    3. The attempt at a solution

    Theorem:
    If [itex]L:U\rightarrow V[/itex] is a linear transformation and [itex]\dim U=n[/itex], then [itex]\dim Ker L+\dim C(L^T)=n[/itex]. [itex]Ker L[/itex] is the null space, [itex]C(L^T)[/itex] is the row space of [itex]L[/itex] and [itex]n[/itex] is the number of column vectors in [itex][L][/itex].

    How to use this theorem to prove the given statement?
     
  2. jcsd
  3. Feb 22, 2016 #2

    fresh_42

    Staff: Mentor

    You may consider [itex]U=\mathbb{F}+\ker L[/itex] and ##L: U → V##. Then according to your given formula [itex]\dim U = \dim \ker L + \dim C(L^T)[/itex]. With ##\dim C(L^T) = \dim L(U) = \dim L(\mathbb{F} + \ker L) = \dim L(\mathbb{F})## we have ##\dim(\mathbb{F}+\ker L) = \dim U = \dim \ker L + \dim L(\mathbb{F})##.
     
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