18.700 Problem Set 2 Solutions
1. (3 points) Let V be the vector space of polynomials of degree at most ve with
real coecients. Dene a linear map
T : V R3 ,
T (p) = (p(1), p(2), p(3).
That is, the coordinates of the vector T (p) are the values of p at 1,

18.700 Problem Set 1
Due in class Tuesday September 16; late work will not be accepted. Your work on
graded problem sets should written entirely on your own, although you may consult
others before writing. I have written a few words about what kind of sol

18.700 Problem Set 6
Due in class Tuesday November 5; 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.
1. (8 points) Suppose that we are given three po

18.700 Problem Set 9 solutions
1. (8 points) Suppose V is a real or complex inner product space.
A linear map S L(V ) is called skew-adjoint if S = S. Suppose V is
complex and nite-dimensional, and S is skew-adjoint. Show that the
eigenvalues of S are all

18.700 Problem Set 6 Solutions
1. (8 points) Suppose that we are given three polynomials
p2 (x) = ax2 + bx + c,
p1 (x) = dx + e,
p0 (x) = f
with real coecients. This problem is about the dierential operator
D = p2 (x)
d2
d
+ p1 (x)
+ p0 (x).
2
dx
dx
a) Ex

18.700 Problem Set 8 solutions
1. (12 points) Suppose that V is a (real or complex) inner product
space, and that (t1 , . . . , tn ) is a basis of V .
a) Show that there is just one n n matrix U = (uij ) with the following
properties:
i) U is upper triang

18.700 Problem Set 7 Solutions
1. (6 points) Suppose that V is a complex inner product space with
orthogonal basis (f1 , . . . , fn ), and T L(V ).
a) Prove that any vector v V can be written
n
v=
i=1
v, fi
fi .
fi , fi
This essentially appears in the not

18.700 Problem Set 8
Due in class Tuesday November 25; 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.
1. (12 points) Suppose that V is a (real or com

18.700 Problem Set 9
Due in class Tuesday December 2; 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.
1. (8 points) Suppose V is a real or complex inn

18.700 Problem Set 7
Due in class Thursday November 13; 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.
1. (6 points) Suppose that V is a complex inne

18.700 Problem Set 4 solutions
For the rst problems, you may use the theorem I stated in class Tuesday October
7: there is a one-to-one correspondence
U Row(A)
between r-dimensional subspaces of F n and r n reduced row-echelon matrices
having exactly one

18.700 Problem Set 5 solutions
What I did discuss in class on 10/23 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 o

18.700 Problem Set 6
Due in class Tuesday November 4; 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.
1. (8 points) Suppose that we are given three po

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 pro

18.700 Problem Set 3
Due in class Tuesday October 7; late work will not be accepted. Your work on
graded problem sets should written entirely on your own, although you may consult
others before writing.
1. (3 points) Give an example of a 3 3 matrix A of r

18.700 Problem Set 4
Due in class TUESDAY October 14; 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.
For the rst problems, you may use the theorem I

18.700 Problem Set 2
Due in class Tuesday September 23; late work will not be accepted. Your work
on graded problem sets should written entirely on your own, although you may
consult others before writing.
1. (3 points) Let V be the vector space of polyno

18.700 Problem Set 7
Due in class Tuesday November 12; 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.
1. (6 points) Suppose that V is a complex inner