Linear Algebra

Section VI. Projection 2613.15 What is the orthogonal projection into the trivial subspace?3.16 What is the projection of ⃗v into M along N if ⃗v ∈ M?3.17 Show that if M ⊆ R n is a subspace with orthonormal basis 〈⃗κ 1 , . . . , ⃗κ n 〉 thenthe orthogonal projection of ⃗v into M is this.(⃗v ⃗κ 1 ) · ⃗κ 1 + · · · + (⃗v ⃗κ n ) · ⃗κ ň 3.18 Prove that the map p: V → V is the projection into M along N if and onlyif the map id − p is the projection into N along M. (Recall the definition of thedifference of two maps: (id − p) (⃗v) = id(⃗v) − p(⃗v) = ⃗v − p(⃗v).)̌ 3.19 Show that if a vector is perpendicular to every vector in a set then it isperpendicular to every vector in the span of that set.3.20 True or false: the intersection of a subspace and its orthogonal complement istrivial.3.21 Show that the dimensions of orthogonal complements add to the dimensionof the entire space.̌ 3.22 Suppose that ⃗v 1, ⃗v 2 ∈ R n are such that for all complements M, N ⊆ R n , theprojections of ⃗v 1 and ⃗v 2 into M along N are equal. Must ⃗v 1 equal ⃗v 2 ? (If so, whatif we relax the condition to: all orthogonal projections of the two are equal?)̌ 3.23 Let M, N be subspaces of R n . The perp operator acts on subspaces; we canask how it interacts with other such operations.(a) Show that two perps cancel: (M ⊥ ) ⊥ = M.(b) Prove that M ⊆ N implies that N ⊥ ⊆ M ⊥ .(c) Show that (M + N) ⊥ = M ⊥ ∩ N ⊥ .̌ 3.24 The material in this subsection allows us to express a geometric relationshipthat we have not yet seen between the rangespace and the nullspace of a linearmap.(a) Represent f : R 3 → R given by( )v1v 2 ↦→ 1v 1 + 2v 2 + 3v 3v 3with respect to the standard bases and show that( ) 123is a member of the perp of the nullspace. Prove that N (f) ⊥ is equal to thespan of this vector.(b) Generalize that to apply to any f : R n → R.(c) Represent f : R 3 → R 2 ( )v1( )1v1 + 2v 2 + 3v 3v 2 ↦→4v 1 + 5v 2 + 6v 3v 3with respect to the standard bases and show that( ) ( ) 1 42 , 53 6are both members of the perp of the nullspace. Prove that N (f) ⊥ is the spanof these two. (Hint. See the third item of Exercise 23.)