Linear Algebra

252 Chapter Three. Maps Between Spaces(b) Show that if ⃗v is not a member of the line then the set {⃗v,⃗v − proj [⃗s ] (⃗v )} islinearly independent.1.13 Definition 1.1 requires that ⃗s be nonzero. Why? What is the right definitionof the orthogonal projection of a vector into the (degenerate) line spanned by thezero vector?1.14 Are all vectors the projection of some other vector into some line?̌ 1.15 Show that the projection of ⃗v into the line spanned by ⃗s has length equal tothe absolute value of the number ⃗v • ⃗s divided by the length of the vector ⃗s .1.16 Find the formula for the distance from a point to a line.1.17 Find the scalar c such that the point (cs 1 , cs 2 ) is a minimum distance from thepoint (v 1 , v 2 ) by using calculus (i.e., consider the distance function, set the firstderivative equal to zero, and solve). Generalize to R n .̌ 1.18 Prove that the orthogonal projection of a vector into a line is shorter than thevector.̌ 1.19 Show that the definition of orthogonal projection into a line does not dependon the spanning vector: if ⃗s is a nonzero multiple of ⃗q then (⃗v • ⃗s/⃗s • ⃗s ) · ⃗s equals(⃗v • ⃗q/⃗q • ⃗q ) · ⃗q.̌ 1.20 Consider the function mapping the plane to itself that takes a vector to itsprojection into the line y = x. These two each show that the map is linear, the firstone in a way that is coordinate-bound (that is, it fixes a basis and then computes)and the second in a way that is more conceptual.(a) Produce a matrix that describes the function’s action.(b) Show that we can obtain this map by first rotating everything in the planeπ/4 radians clockwise, then projecting into the x-axis, and then rotating π/4 radianscounterclockwise.1.21 For ⃗a, ⃗b ∈ R n let ⃗v 1 be the projection of ⃗a into the line spanned by ⃗b, let ⃗v 2 bethe projection of ⃗v 1 into the line spanned by ⃗a, let ⃗v 3 be the projection of ⃗v 2 intothe line spanned by ⃗b, etc., back and forth between the spans of ⃗a and ⃗b. That is,⃗v i+1 is the projection of ⃗v i into the span of ⃗a if i + 1 is even, and into the spanof ⃗b if i + 1 is odd. Must that sequence of vectors eventually settle down — mustthere be a sufficiently large i such that ⃗v i+2 equals ⃗v i and ⃗v i+3 equals ⃗v i+1 ? If so,what is the earliest such i?VI.2Gram-Schmidt OrthogonalizationThis subsection is optional. We only need the work done here in the finaltwo sections of Chapter Five. Also, this subsection requires material fromthe previous subsection, which itself was optional.The prior subsection suggests that projecting into the line spanned by ⃗sdecomposes a vector ⃗v into two parts⃗v⃗v − proj [⃗s] (⃗p)proj [⃗s] (⃗p)⃗v = proj [⃗s ] (⃗v) + ( ⃗v − proj [⃗s ] (⃗v) )