Math 212 Final Exam Solutions
(1) (a) Let B denote the standard basis, and B
20
2 1
[T ]B =
1 2
= cfw_v1 , v2 , v3 . Then by denition:
1
0
1
(b) By matrix multiplication:
3
2
[T v1 ]B = 2 [T v2 ]B = 1
0
1
1
[T v3 ]B = 1
3
Using the fact that:
a+b
[av1 + b
Solutions to Exam 2, Math 212 Spring 2012
1. (a) For h(x, y) = 100 x6 4y 2 , h(x, y) = (hx , hy ) = (6x5 , 8y).
(b) The horizontal direction of the initial downhill roll is given by the direction
of h(1, 1) = (6, 8) = (6, 8). Moreover, the length of the g
Solutions to Third Midterm Exam, Math 212 Spring 2012
1. What is the length of the path (t) = (2 cos t, 2 sin t, 2 t3/2 ) for 0 t 5?
3
Since (t) = (2 sin t, 2 cos t, t1/2 )
(t)
=
4 sin2 t + 4 cos2 t + t =
4+t ,
and the length is
5
5
(t) dt =
0
4+5
4 + t
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Math 212 Spring 2005 Exam #2
Instructions: You Have two Hours to complete this exam. You should worE alone, without
access to the textbook or class notes. You may not use a calculator. Do not discuss this exam
with anyone except your instructor.
This exam
Math 212 Spring 2006
Exam #1: Solutions
1. Let P = (2, 3, 1), Q = (2, 2, 2), R = (3, 3, 1).
(a) Find the equation of the plane through P, Q and R.
Solution: We see that the plane is parallel to the vectors v = Q P = (0, 1, 1) and
w = R P = (1, 0, 2) hence
Math 212 Spring 2008 Exam 2
Instructor: S. Cautis
Instructions: This is a closed book, closed notes exam. Use of calculators is
not permitted. You have two hours. Do all 8 problems. You must show your
work to receive full credit on a problem. An answer wi
Solutions, Exam 1
Exercise 1. (a) Since u = 1 + 4 + 4 = 3 and v = 1 + 0 + 1 = 2, the normalized
vectors are
1 2 2
v
1
1
u
=
, ,
and
= , 0, .
u
3 3 3
v
2
2
(b) Since
cos =
uv
u v
=
1+2
1
= , =
.
4
3 2
2
(c) A vector orthogonal to both u and v is u v = (2,
Math 212 Spring 2006
Exam #2 Solutions
Instructions: This was a 2 hour, closed notes, closed book, and pledged exam.
1. Find and classify all the critical points of
1
1
f (x, y) = x2 xy + y 3 .
2
3
Solution: Since f is dierentiable everywhere, the only cr
Interpolation and Approximation
Interpolation is a technique of estimating the value of a
function for any intermediate values of the independent
variable x, given a table of discrete data points (x i , fi),
i = 0,1,n.
Uses of interpolation:
Reconstructin
Math 212 Quiz # 1 Solutions February 11, 2005
124
(1) Find the inverse of the matrix: 0 1 2 Solution. Find elementary row operations
101
1 2 0
to bring the matrix to reduced row echelon form. The inverse is: 2 3 2
1 2
1
(2) Recall that F2 is the eld with
Math 212 Quiz # 2 February 25, 2005
(1) Let W1 and W2 be subspaces of a vector space V such that W1 + W2 = V and
W1 W2 = cfw_0. Prove that for each vector v V there are unique vectors w1 W1
and w2 W2 such that v = w1 + w2 . Solution. By denition of the su
Math 212 Quiz # 3 March 23, 2005
(1) Suppose that T : R3 R3 is a linear map given by the matrix:
1 21
[T ]B = 0 1 1
1 3 4
in the standard basis B . Find a basis for the kernel and range of T . Solution. Find
the reduced row echelon form of [T ]B :
1 0 1
r
Math 212 Quiz # 4 April 8, 2005
(1) True or false: A nonzero polynomial f (x) over a eld K always has a root, i.e. some
c K such that f (c) = 0. Justify your answer. Solution. False. Just consider
f (x) = x2 + 1 over the real numbers.
(2) Give the denitio
Math 212 Quiz # 5 April 22, 2005
(1) Write down both the cofactor and permutation expansions of the determinant of an
n n matrix. Explain your notation. Solution.
n
(1)i+j Aij det Mij
det A =
j =1
sgn( )A1(1) An(n)
det A
Sn
where A = (Aij ) is the matrix
Math 212 Quiz # 6 May 4, 2005
(1) State the Cayley-Hamilton Theorem. Solution. If T : V V is a linear transformation of a nite dimensional vector, and p(x) is the characteristic polynomial of T ,
then p(T ) = 0.
(2) Answer the following True or False. Be
Math 212 Quiz # 5 April 22, 2005
(1) Write down both the cofactor and permutation expansions of the determinant of an
n n matrix. Explain your notation. Solution.
n
(1)i+j Aij det Mij
det A =
j =1
sgn( )A1(1) An(n)
det A
Sn
where A = (Aij ) is the matrix
Honors Linear Algebra, 110.212, Fall 2013
W. Stephen Wilson
W Stephen Wilson email: wsw@math.jhu.edu
Textbook: Linear Algebra, 4th edition, Friedberg, Insel and Spence. ISBN 0-13-008451-4
Monday, Wednesday, 1:30-2:45, Krieger 308 Section, Friday, 1:30, Kr