Week 1 Homework
1. Let V be a vector space of dimension n, and let v1 , . . . , vk be linearly independent vectors in
V , with k < n. Show that there exist vectors vk+1 , . . . , vn such that v1 , . . . , vn is a basis for V .
Solution. Every basis for V
Midterm 2 Solutions
Choose 6 out of 8 problems below.
1. Consider the sets described in (a)-(c) below. For each set, either prove it is an (n1)-manifold
in Rn or give an example to show that it is not.
(a) The graph of a dierentiable function f : U R dene
Midterm 1 Solutions
Choose 6 out of 9 problems below.
1. Assume L : R3 R3 is a linear map such that L(e1 ) = (1, 1, 0), L(e2 ) = (0, 0, 1), and
L(e3 ) = (1, 1, 1). Give bases for Im L and Ker L.
Solution. The image of L is spanned by L(e1 ), L(e2 ), and L
Week 12 Homework
1. Assume f : Rn Rn is a C 1 function with a C 1 inverse locally near a. That is, there is a
neighborhood U of a and a C 1 function g : f (U ) U such that g(f (x) = x for x U . Show
that then f (a) is nonsingular.
Solution. Observe that g
Week 14 Homework
1. If A Rm and B Rn are both contented sets, then A B is contented and v(A B) =
v(A)v(B). Prove this in the following two ways:
(a) Using the denition of volume;
(b) Using Fubinis theorem for the function AB : Rm Rn R.
Recall that if I an
Week 11 Homework
1. Let G : R2 R be a C 1 function such that G(a, b) = 0 and D2 G(a, b) = 0. Then the implicit
function theorem yields a C 1 function f such that the graph of y = f (x) agrees with the zero
set of G in a neighborhood of (a, b). Show that
f
Week 8 Homework
1. Consider the function
f (x) = ex 1
tan1 (x) x .
Show that the 4th order Taylor expansion of f around 0 is
1
f (x) = x4 + R4 (x).
3
(Hint: First compute the Taylor expansions of ex and tan1 around zero.) Observe that 0 is
a critical poin
Week 10 Homework
1. Show that 2 x sin x = 0 has exactly one solution, x , in [/6, /2]. Then show that
(x) = 2 sin x is a contraction mapping on [/6, /2], and calculate x to 3 digits of precision.
Solution. Let f (x) = 2 x sin x. Observe that f (/6) = 2 /6
Week 3 Homework
1. For C, D Rn dene d(C, D) = inf xC, yD |x y|. If C is compact and D is closed, prove
there exist c C and d D such that |c d| = d(C, D).
Hint: First show that the statement is true when C = cfw_a is a single point.
Remark: Below we write
Week 5 Homework
1. Find the shortest distance from the point (1, 0) to the parabola y 2 = 4x.
Solution. The graph of y 2 = 4x is a closed set so there is a minimum distance. Let f (x, y) =
(x 1)2 + y 2 and g(x, y) = y 2 4x; we must minimize f on the zero
Week 4 Homework
1. Let f : Rn R be dierentiable, and assume
()
f (tx) = tf (x)
for every x Rn and t R
(i) Show that1 f (x) = f (0) x, so that f is linear.
(ii) Assume g : Rn R satises (*) but is not linear (i.e. not additive). Show that g has
directional
Week 7 Homework
1. Show that the maximum value of f (x) = x2 x2 . . . x2 on the sphere S n1 = cfw_x Rn : |x| = 1
n
1 2
is (1/n)n . Use this to prove the arithmetic/geometric mean inequality:
n
a1 a2 . . . an
a1 + a2 + . . . + an
n
for positive real numbe
Week 2 Homework
1. Without using any theorems about determinants, prove that if A and B are n n matrices
such that AB = I, then BA = I.
Hint: Note that (I BA)B = 0. What is the kernel of B? Of I BA?
Solution. Let A and B are n n matrices such that AB = I.
Midterm 3
Complete exactly 6 of the 8 problems below. Only 6 will be graded. Indicate which
problems you have chosen by circling the problem number.
1
1. Recall Newtons method for nding roots of a function f :
xn+1 = xn
f (xn )
.
f (xn )
Consider f (x) =