Math 206, Spring 2016
Midterm 3
April 28, 2016
MIDTERM 3
• Complete the following problems. You may use any result from class you like, but if you cite a theorem
be sure to verify the hypotheses are satisfied.
• This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted.
• In order to receive full credit, please show all of your work and justify your answers. You do not need
to simplify your answers unless specifically instructed to do so.
• If you need extra room, use the back sides of each page. If you must use extra paper, make sure to
write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.
Name:
The following boxes are strictly for grading purposes. Please do not mark.
1
16 pts
2
10 pts
3
12 pts
4
12 pts
5
16 pts
6
18 pts
7
16 pts
Total
100 pts
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 1 of 8
Math 206, Spring 2016
Midterm 3
April 28, 2016
(1) (16 pts) Provide the following definitions and theorem statements. Be sure to appropriately quantify
any objects in your definitions.
(a) Suppose that V is a vector space with operations ⊕ and . Carefully write down the two vector
space axioms that can be described as distributive properties.
(b) Define “least squares solution.”
(c) Define “inner product.”
(d) Define “eigenvalue.”
(e) Define “orthogonal transformation.”
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 2 of 8
Math 206, Spring 2016
Midterm 3
April 28, 2016
(2) (10 pts) For each of the following, provide the desired example. Make sure your examples are as specific
as possible. (I.e., don’t attempt to find all examples; just give me one.)
(a) Two distinct vector spaces V and W , each isomorphic to R55 . Neither V nor W is allowed to be a
subspace of any Rn .
(b) An inner product space other than Rn under the dot product. [Give the vector space and its
associated inner product.]
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 3 of 8
Math 206, Spring 2016
Midterm 3
April 28, 2016
(3) (12 points) Let T ∈ L(R2×2 , R2 ) be the transformation defined by
a b
a+b
T
=
.
c d
c+d
For the bases B of R2×2 and D of R2 given by
1 −1
1 1
1
B=
,
,
−1
1
2 2
2
0
1
D=
,
,
1
0
0
0
1
,
3
2
4
compute RepD,B (T ).
(4) (12 points) Suppose that T ∈ L(V, W ), that B = {v1 , v2 } is a basis for V , and that D = {w1 , w2 , w3 }
is a basis for W . Suppose you are given that


2 1
RepD,B (T ) =  1 3  .
−1 0
For v = −v1 + 3v2 , express T (v) as a linear combination of {w1 , w2 , w3 }. [Hint: You might start by
writing v as a vector in B-coordinates.]
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 4 of 8
Math 206, Spring 2016
Midterm 3
April 28, 2016
(5) (16 points)

 
  
−4
3 
 2
(a) Use Gram-Schmidt to produce an orthonormal collection from  2  ,  2  ,  0  .


1
4
0

2
(b) Let Q and R be the matrices from the QR-factorization of  2
1
http://palmer.wellesley.edu/~aschultz/w16/math206

−4 3
2 0  . Give the matrix Q−1 .
4 0
Page 5 of 8
Math 206, Spring 2016
Midterm 3



(6) (18 points) For A = 





(a) Let v = 


1
6
−1
0
−1
2
2
−2
2
0
April 28, 2016


 
0
1/2
3 −3




 1/2   2/3
3 −1 

 


 

−3
5 
 , it is a fact that im(A) has orthonormal basis  −1/2  ,  2/3

 1/2   0
3 −3 



1/3
0
0
1



. Compute projim(A) v.


(b) For v from part (a), find vectors vk ∈ im(A) and v⊥ ∈ im(A)⊥ so that v = vk + v⊥ .
(c) What is dim(ker(AT ))?
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 6 of 8






 .





Math 206, Spring 2016
Midterm 3
April 28, 2016
(7) (16 pts) For the following problems, V is a vector space with operations ⊕ and . Carefully apply vector
space axioms or theorems we discussed in class or on homework to prove the following statements. All
steps should have accompanying justification.
(a) Suppose that v ∈ V is not the neutral element 0V , and that k ∈ R is nonzero. Prove that
k v 6= 0V . [Hint: you might try contradiction. You may use familiar rules of real arithmetic.]
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 7 of 8
Math 206, Spring 2016
Midterm 3
April 28, 2016
(b) Suppose that v ∈ V is not the neutral element 0V , and let −v ∈ V be the vector satisfying
v ⊕ (−v) = 0V . (I.e., −v is the additive inverse of v.) Prove that v 6= −v. In addition to vector
space axioms and theorems from class and homework, you may also use the result from part (a) if
you like (even if you don’t succeed in proving (a)). [Hint: you might try contradiction.]
http://palmer.wellesley.edu/~aschultz/w16/math206
Page 8 of 8