Linear Algebra

Section II. Linear Geometry of n-Space 43⎛ ⎞( ( ) ( ) ( ) 14 0 3 −1⎜−1⎟(a) (b)(c) 1 (d) 0 (e)1)2⎝1⎠100̌ 2.11 Find the angle between each two, if it is defined.( ( ( ) ( ) 1 0 ( ( ) 1 1 1 1(a) , (b) 2 , 4 (c) , 42)4)2)0 1−1̌ 2.12 During maneuvers preceding the Battle of Jutland, the British battle cruiserLion moved as follows (in nautical miles): 1.2 miles north, 6.1 miles 38 degreeseast of south, 4.0 miles at 89 degrees east of north, and 6.5 miles at 31 degreeseast of north. Find the distance between starting and ending positions. [Ohanian]2.13 Find k so that these two vectors are perpendicular.( ) (k 41 3)2.14 Describe the set of vectors in R 3 orthogonal to this one.( ) 13−1̌ 2.15 (a) Find the angle between the diagonal of the unit square in R 2 and one ofthe axes.(b) Find the angle between the diagonal of the unit cube in R 3 and one of theaxes.(c) Find the angle between the diagonal of the unit cube in R n and one of theaxes.(d) What is the limit, as n goes to ∞, of the angle between the diagonal of theunit cube in R n and one of the axes?2.16 Is any vector perpendicular to itself?̌ 2.17 Describe the algebraic properties of dot product.(a) Is it right-distributive over addition: (⃗u + ⃗v) ⃗w = ⃗u ⃗w + ⃗v ⃗w?(b) Is is left-distributive (over addition)?(c) Does it commute?(d) Associate?(e) How does it interact with scalar multiplication?As always, any assertion must be backed by either a proof or an example.2.18 Verify the equality condition in Corollary 2.6, the Cauchy-Schwartz Inequality.(a) Show that if ⃗u is a negative scalar multiple of ⃗v then ⃗u ⃗v and ⃗v ⃗u are lessthan or equal to zero.(b) Show that |⃗u ⃗v| = ‖⃗u ‖ ‖⃗v ‖ if and only if one vector is a scalar multiple ofthe other.2.19 Suppose that ⃗u ⃗v = ⃗u ⃗w and ⃗u ≠ ⃗0. Must ⃗v = ⃗w?̌ 2.20 Does any vector have length zero except a zero vector? (If “yes”, produce anexample. If “no”, prove it.)̌ 2.21 Find the midpoint of the line segment connecting (x 1 , y 1 ) with (x 2 , y 2 ) in R 2 .Generalize to R n .2.22 Show that if ⃗v ≠ ⃗0 then ⃗v/‖⃗v ‖ has length one. What if ⃗v = ⃗0?2.23 Show that if r ≥ 0 then r⃗v is r times as long as ⃗v. What if r < 0?