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Kernel subsets of transformations

  1. Dec 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Let [itex] T_1,T_2:ℝ^n\rightarrowℝ^n [/itex] be linear transformations. Show that [itex] \exists S:ℝ^n\rightarrowℝ^n [/itex] s.t. [itex] T_1=S\circ T_2 \Longleftrightarrow kerT_2\subset kerT_1 [/itex].

    3. The attempt at a solution

    [itex] (\Longrightarrow) [/itex] Let [itex] S:ℝ^n\rightarrowℝ^n [/itex] be a linear transformation s.t. [itex] T_1 = S\circ T_2 [/itex] and let [itex]\vec{v}\in kerT_2 [/itex]. Then [itex] S(T_2(\vec{v})) = S(\vec{0}) = \vec{0} [/itex] by linearity. Then [itex]T_1(\vec{v}) = \vec{0}[/itex]. Thus [itex] \vec{v}\in kerT_1 \quad \forall\vec{v}\in kerT_2 [/itex]. Therefore [itex]kerT_2 \subset kerT_1 [/itex].

    [itex] (\Longleftarrow) [/itex] Suppose that [itex] kerT_2\subset kerT_1[/itex] and choose [itex]S:ℝ^n\rightarrowℝ^n[/itex] s.t. [itex]S[/itex] is linear and [itex]T_1 = S\circ T_2 [/itex]. Then for [itex]\vec{v}\in kerT_2,\quad T_1(\vec{v}) = S(T_2(\vec{v}) = S(\vec{0}) = \vec{0}. [/itex] Thus there exists such a transformation.
     
  2. jcsd
  3. Dec 12, 2012 #2

    Dick

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    The first part seems ok. For the second, the problem is to show that such an S exists given ker(T2) is contained in ker(T1). Not to assume it exists.
     
    Last edited: Dec 12, 2012
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