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Exercise 23

Avg difficulty 3.0

Consider how ‘dimension’ interacts with ‘subset’. Assume UU and WW are both subspaces of some vector space, and that U⊆WU\subseteq W.

  • Prove that dim⁡(U)≤dim⁡(W)\dim(U)\leq\dim(W).

  • Prove that equality of dimension holds if and only if U=WU=W.

  • Show that the prior item does not hold if they are infinite-dimensional.

Hefferon, Linear Algebra, Vector Spaces, Dimension, Exercise 23

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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.