Linear Algebra

Section VI. Projection 251the sub’s shifted position.( ) ( )0 1• ( ) ( )−2 3 1 −3/5⃗p = ( ) ( ) · =1 1 3 −9/5•3 3The distance between ⃗v and ⃗p is about 0.63 miles. The ship will not be in range.This subsection has developed a natural projection map, orthogonal projectioninto a line. As suggested by the examples, we use it often in applications.The next subsection shows how the definition of orthogonal projection into a linegives us a way to calculate especially convenient bases for vector spaces, againsomething that we often see in applications. The final subsection completelygeneralizes projection, orthogonal or not, into any subspace at all.Exerciseš 1.7 Project the first vector orthogonally into the line spanned by the second vector.⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞( ( ) ( ( 1 11 32 3 2 3(a) , (b) , (c) ⎝1⎠,⎝ 2⎠(d) ⎝1⎠,⎝ 3⎠1)−2 1)0)4 −14 12̌ 1.8 Project the vector orthogonally into the line.⎛ ⎞ ⎛ ⎞2 −3(a) ⎝−1⎠ , {c ⎝ 1⎠ ∣ ( )−1c ∈ R} (b) , the line y = 3x−14 −31.9 Although pictures guided our development of Definition 1.1, we are not restrictedto spaces that we can draw. In R 4 project this vector into this line.⎛ ⎞⎛ ⎞1−1⃗v = ⎜2⎟⎝1⎠ l = {c · ⎜ 1⎟ ∣⎝−1⎠c ∈ R}31̌ 1.10 Definition 1.1 uses two vectors ⃗s and ⃗v. Consider the transformation of R 2resulting from fixing( 3⃗s =1)Apply it to these vec-and projecting ⃗v into the line that is the span of ⃗s.tors. ( ( 1 0(a) (b)2)4)Show that in general the projection transformation is this.( ) ( )x1 (x1 + 3x 2 )/10↦→x 2 (3x 1 + 9x 2 )/10Express the action of this transformation with a matrix.1.11 Example 1.5 suggests that projection breaks ⃗v into two parts, proj [⃗s ] (⃗v ) and⃗v − proj [⃗s ] (⃗v ), that are non-interacting. Recall that the two are orthogonal. Showthat any two nonzero orthogonal vectors make up a linearly independent set.1.12 (a) What is the orthogonal projection of ⃗v into a line if ⃗v is a member of thatline?