Linear Algebra, 2020a

Section VI. Projection 279
(a)
( 1
2)
(b)
( 0
4)
Show that in general the projection transformation is this.
( ) ( )
x1 (9x1 + 3x 2 )/10
↦→
x 2 (3x 1 + x 2 )/10
Express the action of this transformation with a matrix.
1.10 Example 1.4 suggests that projection breaks ⃗v into two parts, proj [⃗s ] (⃗v ) and
⃗v − proj [⃗s ] (⃗v ), that are non-interacting. Recall that the two are orthogonal. Show
that any two nonzero orthogonal vectors make up a linearly independent set.
1.11 (a) What is the orthogonal projection of ⃗v into a line if ⃗v is a member of that
line?
(b) Show that if ⃗v is not a member of the line then the set {⃗v,⃗v − proj [⃗s ] (⃗v )} is
linearly independent.
1.12 Definition 1.1 requires that ⃗s be nonzero. Why? What is the right definition
of the orthogonal projection of a vector into the (degenerate) line spanned by the
zero vector?
1.13 Are all vectors the projection of some other vector into some line?
1.14 Show that the projection of ⃗v into the line spanned by ⃗s has length equal to
the absolute value of the number ⃗v • ⃗s divided by the length of the vector ⃗s .
1.15 Find the formula for the distance from a point to a line.
1.16 Find the scalar c such that the point (cs 1 ,cs 2 ) is a minimum distance from the
point (v 1 ,v 2 ) by using Calculus (i.e., consider the distance function, set the first
derivative equal to zero, and solve). Generalize to R n .
̌ 1.17 Let ⃗p be the orthogonal projection of ⃗v ∈ R n onto a line l. Show that ⃗p is the
point in the line closest to ⃗v.
1.18 Prove that the orthogonal projection of a vector into a line has length less than
or equal to that of the vector.
̌ 1.19 Show that the definition of orthogonal projection into a line does not depend
on 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 its
projection into the line y = x. These two each show that the map is linear, the first
one 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 plane
π/4 radians clockwise, then projecting into the x-axis, and then rotating π/4 radians
counterclockwise.
1.21 For ⃗a, ⃗b ∈ R n let ⃗v 1 be the projection of ⃗a into the line spanned by ⃗b, let ⃗v 2 be
the projection of ⃗v 1 into the line spanned by ⃗a, let ⃗v 3 be the projection of ⃗v 2 into
the 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 span
of ⃗b if i + 1 is odd. Must that sequence of vectors eventually settle down — must
there 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?