Math 337 – Sample Exam
Instructions. This sample exam is provided as an aid for preparing for the first midterm
exam. Additionally, preparation for the midterm exam should include review of lectures,
quizzes and all assigned homework problems.
Problem 1. Suppose that u= (1,2), v= (3,4) and w= (1,4,0). Where possible
compute the following:
(a)u·v(b)u·w(c)(u+v)·(u−v)− ||u||2+||v||2
Problem 2. Suppose that uand vare unit vectors and that u·v<0. Explain why
||u+v|| <√2.
Problem 3. Let u= (1,1) and v= (1,−1). Sketch the set of all the linear combinations
of uand vfor which the coefficients in the linear combination are greater or equal to zero
but less than or equal to one. That is, sketch the set
{αu+βv|0≤α≤1 and 0 ≤β≤1}.
Problem 4. Compute the following matrix products (if possible):
(a)1−1
−1 1 1 1
1 1 (b)a b
c d d−b
−c a
(c) [ 1 2 3 4 ]
4
3
2
1
(d)
4
3
2
1
[ 1 2 3 4 ]
Problem 5. Find the coefficients that show that x= (1,2,3,4) is a linear combination
of y= (1,0,0,0), z= (1,1,0,0), w= (1,1,1,0) and v= (1,1,1,1).
Problem 6. For each equation find the matrix Athat makes the equation true:
(a)A
123
124
125
=
123
001
125
(b)A
123
124
125
=
123
001
002
(c)A
123
124
125
=
123
125
124
