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The algebra of linear transformations

Deciding which identities for sums, scalar multiples, and composites of linear transformations hold in general, and which fail.

Matrix Theory and Linear Algebra, The algebra of linear transformations

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1 exercise

Exercise 10.2.1

Suppose $V$ and $W$ are vector spaces, $T,S:V\to W$ and $R,Q:W\to W$ are linear transformations, and $k$ is a scalar. Which of the following equalities are valid?

$R\circ(T+kS)=R\circ T+kR\circ S$ .
$(R+Q)\circ(R+Q)=R\circ R+2R\circ Q+Q\circ Q$ .
$(R+Q)\circ(T+S)=R\circ T+R\circ S+Q\circ T+Q\circ S$ .
If $T$ , $S$ , and $T+S$ are invertible, then $(T+S)^{-1}=T^{-1}+S^{-1}$ .

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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.