, use the
GramSchmidt process to obtain an orthonormal basis for
3
.
(b)
Using your work from part (a), obtain a QRfactorization of
the matrix
A=
.
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View Full DocumentFinal Exam/MAS3105
Page 6 of 7
5. (10 pts.)
Let
100
A=
012
021
.
(a)
Compute the characteristic polynomial p
A
(
λ
) of A.
Leave the
polynomial in factored form.
(Hint: Expand the required
determinant using the first row or column.
Don’t mess with the
linear factor which is multiplied by the2x2 determinant, which
ends up being a difference of squares.)
p
A
(
λ
)=
(b)
List the eigenvalues of A.
(c)
Obtain a basis, B
λ
, for each eigenspace, E
λ
,o
fA
.
Label correctly.
(d)
Obtain an invertible matrix P and a diagonal matrix D so
that A = PDP
1
.
P=
D=
(e)
Verify your P and D work without finding the inverse of P.
(Hint:
There is a matrix equation that is equivalent to the one
in part (d).
Final Exam/MAS3105
Page 7 of 7
6. (8 pts.)
(a)
Show that
W={
a
0
+a
1
t+a
2
t
2
:a
0
1
2
=0
}
is a subspace of P
2
.
(b)
Obtain a basis for W.
(Hint: Use the coordinate mapping with respect to the natural
basis for P
2
to obtain an equivalent problem in
3
.
Solve that
problem and translate the results to the polynomial space.
Observe that the linear equation you have to solve is the one
giving the defining condition on the members of W.)
7. (7 pts.)
Let T: P
2
→
P
2
be the function defined by the rule
T(p) = p
″
2
p
′
,
where the primes denote differentiation with respect to t.
(a)
Verify that T is a linear transformation. (You need only
quote the appropriate properties of differentiation.)
(b)
Obtain the matrix [T]
B
, whereB={
1
,1+t
,1+t+t
2
}i
s
an off the wall basis for P
2
.
(c)
What can you tell about the dimensions of the kernel and the
range by using the matrix [T]
B
??
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Final Exam/MAS3105

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