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Final Exam, December 19, 2013
Final Exam
Linear Algebra, Dave Bayer, December 19, 2013
[1] Find the intersection of the following two affine subspaces of R3 .
 
 


x
1
1 0 y = 1 + 1 1 a
b
1
0 1
z


 


x
1
1 0 y = 2 + 0 1 c
d
z
2
0 1
Final Exam, December 19, 2013
[2] Find an orthogonal basis for the subspace of R4 defined by the equation w + x − y − z = 0. Extend this
basis to a orthogonal basis for R4 .
Final Exam, December 19, 2013
[3] Find the determinant of the matrix










2
2
0
0
0
0
0
1
2
2
0
0
0
0
0
1
2
2
0
0
0
0
0
1
2
2
0
0
0
0
0
1
2
2
0
0
0
0
0
1
2
2
0
0
0
0
0
1
2










Final Exam, December 19, 2013
[4] Solve the differential equation y 0 = Ay where
2 1
A =
,
3 0
λ = −1, 3
e
At
e−t
=
4
1 −1
−3
3
e 3t
+
4
y(0) =
3 1
3 1
1
0
e−t
y =
4
1
−3
e3t
+
4
3
3
Final Exam, December 19, 2013
[5] Express the quadratic form
− 4xy + 3y2
as a sum of squares of othogonal linear forms.
4
1 4 2
1 −2
+
λ = −1, 4
A =
= −
4
5 2 1
5 −2
4
1
0 −2
x
2
− 4xy + 3y = x y
(x − 2y)2
= − (2x + y)2 +
−2
3
y
5
5
0 −2
−2
3
Final Exam, December 19, 2013
[6] Solve the recurrence relation
f(0) = a,
f(n + 1)
f(n)
=
f(1) = b,
3 −2
1
0
n b
a
f(n) = 3 f(n − 1) − 2 f(n − 2)
=
−1 2
−1 2
b
a
+ 2
f(n) = ( − b + 2a) + 2n (b − a)
n
2 −2
1 −1
b
a
Final Exam, December 19, 2013
[7] Find eAt where A is the matrix


1 2 1
A = 0 2 0
1 2 1

λ = 0, 2, 2
eAt





1 0 −1
1 0 1
0 2 0
2t
1
e 
0 +
0 2 0  + te2t  0 0 0 
=  0 0
2
2
−1 0
1
1 0 1
0 2 0
Final Exam, December 19, 2013
[8] Solve the differential equation y 0 = Ay where


−2 2 −1
A =  −1 1 −2  ,
−1 1
1


2
y(0) =  0 
1
λ = 0, 0, 0





3 −3 −3
1 0 0
−2 2 −1
2
t 
3 −3 −3 
=  0 1 0  + t  −1 1 −2  +
2
0
0
0
0 0 1
−1 1
1

eAt
 



3
2
−5
2
t
3
y =  0  + t  −4  +
2
0
1
−1
