18.700 Problem Set 5
Due in class Tuesday October 28; late work will not be accepted. Your work
on graded problem sets should be written entirely on your own, although you may
consult others before writing.
Because I did not introduce the Gram-Schmidt process in class on Thursday, this
problem set will not cover that. (Nevertheless it would be an excellent idea this
weekend to read the description of the process in the “Notes on orthogonal bases”
on the class web site; you’ll understand much better the quick account in class next
Tuesday.
What I did discuss in class on Thursday was several “simpler” notions of the
“size” of a vector in Rn . Here are three such notions
∥(x1 , . . . , xn )∥1 = |x1 | + |x2 | + · · · + |xn |
(discussed in class);
∥(x1 , . . . , xn )∥∞ = largest of |x1 |, |x2 |, . . . , |xn |
(useful for example if the different coordinates correspond to different kinds of bad
thing, and you want to be sure that all of them are small); and
∥(x1 , . . . , xn )∥2 =
!
(x1 )2 + (x2 )2 + · · · + (xn )2
(the formula familiar from 18.02, but maybe not so easy to understand). The first
homework problems are about these notions of size.
1. (3 points) This problem is about the x-axis L = {(x, 0) | x ∈ R}.
a) Find all the points on L that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥1 size: that is, all the (x, 0) so that ∥(3, 2) − (x, 0)∥1 is as small as
possible.
b) Find all the points on L that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥∞ size.
c) Find all the points on L that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥2 size.
2. (6 points) This problem is about the diagonal line D = {(t, t) | t ∈ R}.
a) Find all the points on D that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥1 size.
b) Find all the points on D that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥∞ size.
c) Find all the points on D that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥2 size.
3. (6 points) This problem is about the line T = {(2s, s) | s ∈ R}.
a) Find all the points on T that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥1 size.
1
2
b) Find all the points on T that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥∞ size.
c) Find all the points on T that are as close as possible to the point (3, 2) in terms
of the ∥ · ∥2 size.
4. (3 points) Suppose r1 , r2 , and r3 are real numbers such that r12 + r22 + r32 = 1.
Given (x, y, z) ∈ R3 , find a formula for the point on the line L = {(tr1 , tr2 , tr3 ) |
t ∈ R} that is closest to (x, y, z) in the ∥ · ∥2 size. (You can find this formula in
many places. You’re also being asked to write an explanation of why it’s true.)
5. (3 points) Suppose we are given
three distinct elements x1 , x2 , and x3 in F ;
(∗)
three arbitrary elements a, b, and c in F ;
(∗∗)
and
The problem is to find all polynomials
p(x) = u0 + u1 x + u2 x2 + u3 x3
(∗ ∗ ∗)
of degree less than or equal to three satisfying the conditions
p(x1 ) = a,
p(x2 ) = b,
p(x3 ) = c.
(∗ ∗ ∗∗)
a) The conditions on p can be written as a system of three simultaneous linear
equations in four unknowns. Write the augmented matrix of this system of
equations.
b) Perform elementary row operations to bring this augmented matrix to reduced
row-echelon form. (This is a bit disconcerting, because some of the entries of the
matrix are not “numbers” like 7, but symbols for numbers, like x2 . You know
from algebra how to add, subtract, and multiply such symbols. What requires
care is dividing: before you divide by something like x2 , you need to explain why
it is not zero, or else worry separately about the case when it is zero. But you
should be able to manage.
c) Write all the polynomials of degree less than or equal to three satisfying the
condition (∗ ∗ ∗∗).