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Back to Projection Into a Subspace

Exercise 17

Avg difficulty 3.0

Let M,NM,N be subspaces of Rn\mathbb{R}^{n}. The perp operator acts on subspaces; we can ask how it interacts with other such operations.

  • Show that two perps cancel: (M⊥)⊥=M(M^{\perp})^{\perp}=M.

  • Prove that M⊆NM\subseteq N implies that N⊥⊆M⊥N^{\perp}\subseteq M^{\perp}.

  • Show that (M+N)⊥=M⊥∩N⊥(M+N)^{\perp}=M^{\perp}\cap N^{\perp}.

Hefferon, Linear Algebra, Maps Between Spaces, Projection Into a Subspace, Exercise 17

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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.