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Exercise 10

Avg difficulty 3.0

This exercise makes the notation of writing ‘++’ between sets more natural. Prove that, where W1,…,WkW_{1},\dots,W_{k} are subspaces of a vector space,

W1+⋯+Wk={w⃗1+w⃗2+⋯+w⃗k∣w⃗1∈W1,…,w⃗k∈Wk},W_{1}+\dots+W_{k}=\text{$\{\hskip 1.00006pt\vec{w}_{1}+\vec{w}_{2}+\dots+\vec{w}_{k}\mid\vec{w}_{1}\in W_{1},\dots,\vec{w}_{k}\in W_{k}\hskip 1.00006pt\}$},

and so the sum of subspaces is the subspace of all sums.

Hefferon, Linear Algebra, Vector Spaces, Combining Subspaces, Exercise 10

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Adapted from Linear Algebra © 2016 Jim Hefferon. Licensed under CC BY-SA 3.0 US. Source: Linear Algebra. Imported at revision df2262e.