Math 121: Linear Algebra and Applications
Solution Set 3
Exercise 1 (1.6/35).
(a) Let v + W be a typical element of V /W . Since v V , we can write it as
v = a1 u1 + . . . + ak uk + ak+1 uk+1 + . . . + an un
Since u1 , . . . uk are all in W , in V /W this
Math 121: Linear Algebra and Applications
Solution Set 2
Exercise 1 (1.4/12).
Recall that for if and only if statements we need to prove both directions: assuming the
statement to the left prove the one to the right, and viceversa.
In our case, suppose th
Solutions
Midterm 1 for Math 121
1. (20 points)
a) True
b) False
c) False
d) True
e) True
2. (40 points)
a) Clearly, the zero vector lies in each of the subsets in (i), (ii). Thus, check if for any two
vectors v, w from the subset and for any R the vector
Midterm 2 for Math 121
Last Name:
First Name:
Apart from question 1, you must fully justify your answers. On this exam the full
score is 100 points.
You may assume all vector spaces are nite-dimensional unless otherwise stated. Recall
that for a linear tr
Math 121: Linear Algebra and Applications
Solution Set 1
Exercise 1.
(a) TRUE. This is one of the vector space axioms (VS 3).
(b) FALSE. We proved this in class but here is a short proof. Suppose there is another 0 vector,
call it 0 . Then for any vector
Math 121: Linear Algebra and Applications
Solution Set 6
Exercise 1 (2.6/1d-h, 2.7/1c-g).
(a) (d) TRUE. Note that for any nite dimensional V , V = (V )
(e) FALSE. Let V = R3 and consider the standard basis = cfw_e1 , e2 , e3 . Its dual basis in
(R3 ) is g
Midterm 1 for Math 121
3 pages (including this one)
Last Name:
First Name:
Apart from question 1, you must fully justify your answers. On this exam the full
score is 100 points.
You may assume all vector spaces are nite-dimensional unless otherwise stated