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Math 225
Spring 2008
G. Rosen
Practice Problems for Second Midterm
1.
Let T:V
→
W be a linear transformation and let {
v
1
,
v
2
,. . . ,
v
n
} be a basis for V.
Show that T is 1 – 1 and onto if and only if {T(
v
1
),
T(
v
2
), . . . ,
T(
v
n
)} is a basis
for W.
2.
Let V be the set of all sequences {a
i
}
i=1,
∞
= { a
1
, a
2
, a
3
, a
4
, . . . } of real numbers.
a)
Show that V is a linear vector space over the reals with addition and scalar
multiplication defined in the usual way.
b)
Show that V must be infinite dimensional by showing that it contains an
infinite set of linearly independent vectors.
c)
Let L: V
→
V be the left shift operator defined by
L({a
i
}
i=1,
∞
) = L({ a
1
, a
2
, a
3
, a
4
, . . . }) = { a
2
, a
3
, a
4
, . . . },
and show that L is a linear transformation of V into itself.
d)
Show that L is
NOT
11, find the kernel of L, Ker(L), what is dim(Ker(L)),
and find a basis for Ker(L).
e)
Show that L is onto V.
3.
Given the matrix
A
010
100
001
A
⎡
⎤
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣
⎦
a. Find the eigenvalues of A.
b. Find an
orthonormal eigenbasis
(with repect to the matrix A) for R
3
c. Find a similarity transform which diagonalizes A. (Helpful Time Saving Hint:
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This note was uploaded on 03/31/2008 for the course MATH 225 taught by Professor Guralnick during the Spring '07 term at USC.
 Spring '07
 Guralnick
 Math, Differential Equations, Linear Algebra, Algebra, Equations

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