Linear Algebra

44 Chapter One. Linear Systemš 2.24 A vector ⃗v ∈ R n of length one is a unit vector. Show that the dot productof two unit vectors has absolute value less than or equal to one. Can ‘less than’happen? Can ‘equal to’?2.25 Prove that ‖⃗u + ⃗v ‖ 2 + ‖⃗u − ⃗v ‖ 2 = 2‖⃗u ‖ 2 + 2‖⃗v ‖ 2 .2.26 Show that if ⃗x ⃗y = 0 for every ⃗y then ⃗x = ⃗0.2.27 Is ‖⃗u 1 + · · · + ⃗u n‖ ≤ ‖⃗u 1‖ + · · · + ‖⃗u n‖? If it is true then it would generalizethe Triangle Inequality.2.28 What is the ratio between the sides in the Cauchy-Schwartz inequality?2.29 Why is the zero vector defined to be perpendicular to every vector?2.30 Describe the angle between two vectors in R 1 .2.31 Give a simple necessary and sufficient condition to determine whether theangle between two vectors is acute, right, or obtuse.̌ 2.32 Generalize to R n the converse of the Pythagorean Theorem, that if ⃗u and ⃗vare perpendicular then ‖⃗u + ⃗v ‖ 2 = ‖⃗u ‖ 2 + ‖⃗v ‖ 2 .2.33 Show that ‖⃗u ‖ = ‖⃗v ‖ if and only if ⃗u + ⃗v and ⃗u − ⃗v are perpendicular. Givean example in R 2 .2.34 Show that if a vector is perpendicular to each of two others then it is perpendicularto each vector in the plane they generate. (Remark. They could generatea degenerate plane — a line or a point — but the statement remains true.)2.35 Prove that, where ⃗u, ⃗v ∈ R n are nonzero vectors, the vector⃗u‖⃗u ‖ + ⃗v‖⃗v ‖bisects the angle between them. Illustrate in R 2 .2.36 Verify that the definition of angle is dimensionally correct: (1) if k > 0 thenthe cosine of the angle between k⃗u and ⃗v equals the cosine of the angle between⃗u and ⃗v, and (2) if k < 0 then the cosine of the angle between k⃗u and ⃗v is thenegative of the cosine of the angle between ⃗u and ⃗v.̌ 2.37 Show that the inner product operation is linear: for ⃗u, ⃗v, ⃗w ∈ R n and k, m ∈ R,⃗u (k⃗v + m ⃗w) = k(⃗u ⃗v) + m(⃗u ⃗w).̌ 2.38 The geometric mean of two positive reals x, y is √ xy. It is analogous to thearithmetic mean (x + y)/2. Use the Cauchy-Schwartz inequality to show that thegeometric mean of any x, y ∈ R is less than or equal to the arithmetic mean.? 2.39 A ship is sailing with speed and direction ⃗v 1 ; the wind blows apparently(judging by the vane on the mast) in the direction of a vector ⃗a; on changing thedirection and speed of the ship from ⃗v 1 to ⃗v 2 the apparent wind is in the directionof a vector ⃗ b.Find the vector velocity of the wind. [Am. Math. Mon., Feb. 1933]2.40 Verify the Cauchy-Schwartz inequality by first proving Lagrange’s identity:( ) 2 ( ) ( )∑∑ ∑ ∑a jb j =− (a k b j − a jb k ) 21≤j≤n1≤j≤na 2 j1≤j≤nb 2 j1≤k