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Orthogonal Stuff
David Jekel
February 8, 2016
This worksheet was made for UCLA Math 33A Winter 2016 with Omer Ben Neria; it covers material related to Sections 5.1 and 5.2 of Otto Bretscher’s Linear Algebra with Applications.
Part 1: Gram-Schmidt Computation
Define
~v 1 =
(^) , ~v 2 =
(^) , ~v 3 =
a. Compute ~u 1 = ~v 1 / ‖~v 1 ‖.
b. Compute w~ 2 = ~v 2 − (~v 2 · ~u 1 )~u 1.
c. Make sure w~ 2 is orthogonal to ~u 1. Why does this happen?
d. Compute ~u 2 = w~ 2 / ‖ w~ 2 ‖.
e. Compute w~ 3 = ~v 3 − (~v 3 · ~u 1 )~u 1 − (~v 3 · ~u 2 )~u 2.
f. Verify that w~ 2 is orthogonal to ~u 1 and ~u 2.
g. Compute ~u 3 = w~ 3 / ‖ w~ 3 ‖.
h. Verify directly that ~u 1 , ~u 2 , and ~u 3 are orthonormal (that is, they are unit vectors that are perpendicular to each other). If they are not orthonormal, then check your computations for fraction and square-root errors!
Part 2: Thinking about Gram-Schmidt
a. Show that
span(~u 1 ) = span(~v 1 ) span(~u 1 , ~u 2 ) = span(~v 1 , ~v 2 ) span(~u 1 , ~u 2 , ~u 3 ) = span(~v 1 , ~v 2 , ~v 3 ).
b. What orthonormal basis would you get if you applied Gram-Schmidt to ~v 3 , ~v 2 , ~v 1 (in that order) instead of ~v 1 , ~v 2 , ~v 3?
c. Why are there square roots in the ~uj ’s, but not the w~j ’s?
d. If ~x is a nonzero vector and c is a positive number, then (c~x)/ ‖c~x‖ = ~x/ ‖~x‖. How might this save you time in the Gram-Schmidt computations?
Part 4: Projection
a. Let V = span(~v 1 , ~v 2 ) = span(~u 1 , ~u 2 ). What is the orthogonal projection of
(^) onto V? (See page 206) Compute it for this specific example.
b. If ~x is any vector, show that
(~u 1 · ~x)~u 1 + (~u 2 · ~x)~u 2 = ~u 1 ~uT 1 ~x + ~u 2 ~uT 2 ~x.
c. Let P be the matrix of projection onto V. Show that
P = ~u 1 ~uT 1 + ~u 2 ~uT 2 =
~u 1 ~u 2
~u 1 ~u 2
)T
d. Compute P for this specific example. If you want, verify that the formulas in part (b) and (c) are true for this specific P.
e. Show directly from part (c) that P ~u 1 = ~u 1 , P ~u 2 = ~u 2 , P ~u 3 = 0.
f. Directly from the equations in part (e), show that
P
~u 1 ~u 2 ~u 3
~u 1 ~u 2 ~u 3
and hence
P =
~u 1 ~u 2 ~u 3
~u 1 ~u 2 ~u 3
g. Deduce that all projections onto planes in R^3 are similar to each other.
h. Suppose that some matrix A satisfies A~u 1 = ~u 1 , A~u 2 = ~u 2 , A~u 3 = 0. Show that A also satisfies the equations in part (e) and conclude that A = P.
i. Conclude that P is the unique matrix that satisfies
P ~w = w~ for w~ ∈ V P ~w = ~0 for w~ ∈ V ⊥.
j. How would you change the answers if V were a line instead of a plane? Is the projection onto a line similar to the projection onto a plane?
