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Exercise 10.2.1

Avg difficulty 3.0

Suppose VV and WW are vector spaces, T,S:V→WT,S:V\to W and R,Q:W→WR,Q:W\to W are linear transformations, and kk is a scalar. Which of the following equalities are valid?

  1. R∘(T+kS)=R∘T+kR∘SR\circ(T+kS)=R\circ T+kR\circ S.

  2. (R+Q)∘(R+Q)=R∘R+2R∘Q+Q∘Q(R+Q)\circ(R+Q)=R\circ R+2R\circ Q+Q\circ Q.

  3. (R+Q)∘(T+S)=R∘T+R∘S+Q∘T+Q∘S(R+Q)\circ(T+S)=R\circ T+R\circ S+Q\circ T+Q\circ S.

  4. If TT, SS, and T+ST+S are invertible, then (T+S)−1=T−1+S−1(T+S)^{-1}=T^{-1}+S^{-1}.

Selinger, Matrix Theory and Linear Algebra, Linear transformation of vector spaces, The algebra of linear transformations, Exercise 10.2.1

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.