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Exercise 8

Avg difficulty 3.0

Does Example 4.5 generalize? That is, is this true or false: if a vector space VV has a basis ⟨β⃗1,…,β⃗n⟩\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle then it is the direct sum of the spans of the one-dimensional subspaces V=[{β⃗1}]⊕⋯⊕[{β⃗n}]V=[\text{$\{\hskip 1.00006pt\vec{\beta}_{1}\hskip 1.00006pt\}$}]\oplus\dots\oplus[\text{$\{\hskip 1.00006pt\vec{\beta}_{n}\hskip 1.00006pt\}$}]?

Hefferon, Linear Algebra, Vector Spaces, Combining Subspaces, Exercise 8

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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.