Math 317: Linear Algebra
Exam 1
Fall 2015
Name:
*No notes or electronic devices. You must show all your work to receive full credit. When
justifying your answers, use only those techniques that we learned in class up to Section
2.2. Good luck!
1. Let u = (1, 2, 3) and v = (1, 1, 1).
(a) (10 points) Calculate 3u + 2v and u − v.
(b) (10 points) Calculate (3u + 2v) · (u − v), and calculate cos θ where θ is the
angle between 3u + 2v and u − v. You do not have to simplify your answer.
(c) (10 points) Find a vector w so that w is orthogonal to u.
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Math 317: Linear Algebra
Exam 1
Fall 2015
2. Let A be an n × 2 matrix with columns a1 and a2 in that order.
(a) ( 5 points) Find a vector
x
x= 1 ,
x2
such that Ax = 2a1 + 3a2 .
(b)
i. (10 points) Suppose that Ax = 0 for every x ∈ R2 . Prove that A = 0.
That is, A is the zero matrix.
ii. (5 points) Suppose that B is an n × 2 matrix, and that Ax = Bx for every
x ∈ R2 . Prove that A = B. Hint: Use part(i) to help prove this claim.
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Math 317: Linear Algebra
Exam 1
Fall 2015
3. 25 points, 5 points each.
(a) Find a parametric equation for the line 2x1 − x2 = 0. Write your answer as
span(a) for some vector a.
(b) Let T1 : R2 → R2 be the transformation that projects x onto the line 2x1 −
x2 = 0 in R2 . Prove that T1 is linear.
(c) Give the standard matrix that represents T1 .
(d) Let T2 : R2 → R2 be the linear transformation that reflects x across the line
2x1 − x2 = 0. Give the standard matrix that represents T2 .
(e) Give the standard matrix that represents T1 ◦ T2 .
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Math 317: Linear Algebra
Exam 1
Fall 2015
4. (a) (10 points) Find the reduced row echelon form and the rank of the following matrix.


1 0 0 3
 0 1 1 0 .
−1 2 1 −2


1
(b) (10 points) Find (if possible) a linear combination of the vectors v1 =  0 ,
−1
 
 
 
0
0
3
v2 = 1, v3 = 1, that gives b =  0 . Hint: Use your work in (a) to
2
1
−2
help you with this problem.
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Math 317: Linear Algebra
Exam 1
Fall 2015
5. (15 points, 5 points each) Determine if the following statements are true or false. If
the statement is true, give a brief justification. If the statement is false, provide a
counterexample or explain why the statement is not true.
(a) Suppose that A is a 2 × 2 nonsingular matrix. Then rank(A2 ) = 2.
(b) Suppose that A is an m × n matrix where m < n. The system Ax = b will
possess an infinite number of solutions.
(c) Suppose that A is an m × n matrix such that rank(A) = n. Then Ax = b is
consistent for every b ∈ Rm .
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