Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #18
1. Given two vectorvalued functions, T and R, from Rn to Rm , we can dene the
sum, T + R, of T and R by
(T + R)(v) = T (v) + R(v) for all v Rn .
(a) Verify that, if both T and R are linear, then so i
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #13
a b
c d
space of M(2, 2).
1. Let W =
M(2, 2) d = a and c = b . Prove that W is a sub-
Proof: First, observe that the 2 2 zero matrix O =
0 0
0 0
is in W ; hence,
W is not empty.
a b
; so that
b a
Ne
Math 60. Rumbos
Fall 2014
Solutions to Assignment #17
1. Let f : R2 R2 be a function satisfying
f
1
0
2
, f
3
=
0
1
=
5
1
and f
1
1
=
3
.
2
(a) Show that f cannot be linear.
Solution: If f was linear, then we would have that
f
1
1
1
0
+
0
1
= f
= f
1
+f
0
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #14
a b
M(2, 2) d = a and c = b . It was shown in Probc d
lem 1 in Assignment #13 that C(2, 2) is a subspace of M(2, 2).
1. Let C(2, 2) =
(a) Prove that C(2, 2) = spancfw_I, J, where
1 0
0 1
I=
and J =
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #15
1. Let A be an m n matrix, and cfw_e1 , e2 , . . . , en denote the standard basis in Rn .
(a) Prove that Aej is the j th column of the matrix A.
R1
R2
Solution: Write A = . , where R1 , R2 , .
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #16
1. Dene T : R2 R2 as follows: For each v R2 , T (v) is the reection of the
point determined by the coordinates of v, relative to the standard basis in R2 ,
on the line y = x in R2 . That is, T (v) de
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #12
1
1
1. The vectors v1 = 1 , and v2 = 0 span a twodimensional subspace
2
1
3
in R ; in other words, a plane through the origin. Give two unit vectors which
are orthogonal to each other, and which al
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #19
1. Assume that T : Rn Rm is linear. Prove that T is onetoone if and only if
NT = cfw_0, where NT denotes the null space. or kernel, of T
Solution: Assume that T : Rn Rm is linear and that T is oneto
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #22
1. Let T : R2 R2 denote a linear transformation in R2 . Suppose that v1 and v2
are two eigenvectors of T corresponding to the eigenvalues 1 and 2 , respectively.
Prove that, if 1 = 2 , then the set c
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #21
1. Let R : R2 R2 denote rotation around the origin in the counterclockwise
through an angle . Let B = cfw_v1 , v2 , where
2
1
v1 =
and
1
.
2
v2 =
Give the matrix representation for R relative to B; t
Math 60. Rumbos
Fall 2014
1
Solutions to Assignment #20
1. In this problem and problems (2) and (3) you will be proving the Dimension
Theorem
dim(NT ) + dim(IT ) = n,
(1)
for a linear transformation T : Rn Rm .
Show that if NT = Rn , then T must be the ze
Department of Mathematics
Pomona College
Syllabus for Mathematics 60
Fall 2014
Time and Place:
MWF 11:00 am - 11:50 am MDSL 126
Instructor:
Dr. Adolfo J. Rumbos
Office:
MDSL 106
Phone / e-mail:
ext. 18713 / [email protected]
Office Hours:
appointment
MWF