1. Let F be a field and let f be the linear functional on F 2 defined
by f (x1 , x2 ) = ax1 + bx2 . For each of the following linear operators T , let
g = T t f , and find g(x1 , x2 ).
( Hint: Simply you can apply Theorem 21 on page 112 to solve this
ques
Section 6.2 Characteristic Values
4. Let T be the linear operator on R3 which is represented in the standard
ordered basis by the matrix
9 4 4
8 3 4
16 8 7
Prove that T is diagonalizable by exhibiting a basis for R3 , each vector of
which is a character
Section 8.2 Inner Product Spaces
2. Apply the Gram-Schmidt process to the vectors 1 = (1, 0, 1), 2 =
(1, 0, 1), 3 = (0, 3, 4), to obtain an orthonormal basis for R3 with the standard inner product.
Comments: The Gram-Schmidt process is described in the pr
Math 115AH, Final, June 8, 2011
1. If S is a subset of V , we define S 0 is the set of all elements of V which
vanish on S.
a) If V = R3 and
Find a basis of S 0 .
80 1 0
>
>
1
1
>
<B C B
B C B
S = B 0 C, B 1
>
@ A @
>
>
: 3
1
19
>
>
C>
=
C
C .
A>
>
>
;
Co
Section 6.7 Invariant direct sums
1. Let E be a projection of V and let T be a linear operator on V . Prove
that the range of E is invariant under T if and only if ET E = T E. Prove
that both the range and the null space of E are invariant under T if and
Math 115AH, Final, Dec. 13, 2012
1. In this problem, F = C. Suppose V has an inner product < , > and
that 1 , . . . , n are an orthonormal basis of V . Suppose W also has an inner
product < , > and that
1, . . . ,
m
are an orthonormal basis of W . Suppose
Section 6.4 Invariant subspaces
2
1. Let T be the linear
0 operator
1 on R , the matrix of which in the standard
ordered basis is A = @
1
1
A.
2 2
(a) Prove that the only subspace of R2 invariant under T are R2 and the
zero subspace.
Comments: This is sim
Math 115AH, Final, Dec. 5, 2011
1. Suppose B = cfw_1 , 2 , . . . , n is a basis of V and T : V ! V and 1
is an eigenvector of T with eigenvalue
1.
What is the first column of [T ]B ?
Comments: Easy. You can apply the definition of an eigenvalue-eigenvect
Math 115AH, Final, Dec. 10, 2010
1. If S is a subset of V , we define S 0 is the set of all elements of V which
vanish on S.
a) If V = R3 and
Find a basis of S 0 .
80 1 0
>
>
1
1
>
<B C B
B C B
S = B 2 C, B 0
>
@ A @
>
>
: 3
1
19
>
>
C>
=
C
C .
A>
>
>
;
C
Math 115AH, Final, June 9, 2009
1. a) Let x1 , x2 , . . . . , xn be a set of vectors in V . Define what it means
to say x1 , x2 , . . . , xn are linearly independent.
Comments: A basic definition
b) Suppose x, y, z 2 V . Prove that x, y, z are are linearl
Section 8.3
2. Let T be the linear operator on C2 defined by T "1 = (1 + , 2), T "2 =
(, ). Using the standard inner product, find the matrix of T in the standard
ordered basis. Does T commute with T ?
Comments: A trick here is to apply Corollary on page