F14 11:40 Final Exam Problem 1
Linear Algebra, Dave Bayer
[1] Find the intersection of the following two affine subspaces of R4 .
 
w

x
0 1
0
0 
  = 1
2 0 −1 −1  y 
1
z


 


w
2
2 −1  x
 

0
  = 1 + 0
 r
 y
1
1
0 s
z
2
0
1


w


 x
  = 

 y


z


 
 
w
1
1
 x
1
0
  =   +  t
 y
0
1
z
1
1










 + 







t


F14 11:40 Final Exam Problem 2
Linear Algebra, Dave Bayer
[2] Find the 3 × 3 matrix A that maps the vector (1, 1, 0) to (2, 2, 0), and maps each point on the plane
x + y + z = 0 to itself.


A=


3 1 1
1
A = 1 3 1
2
0 0 2
1






F14 11:40 Final Exam Problem 3
Linear Algebra, Dave Bayer
[3] Find the inverse of the matrix


1 0 2
A=1 0 1
1 3 2
A−1 =


−3
6 0
1
−1
0 1
A =
3
3 −3 0
1








F14 11:40 Final Exam Problem 4
Linear Algebra, Dave Bayer
[4] Find An where A is the matrix
A =
0
1
2 −1
An =
λ = −2, 1
A
n
(−2)n
=
3
n 
 +

1 −1
−2
2

1
+
3
2 1
2 1
n 



F14 11:40 Final Exam Problem 5
Linear Algebra, Dave Bayer
[5] Solve the differential equation y 0 = Ay where
2 3
A =
,
1 0
y(0) =
y =
λ = −1, 3
e
At
e−t
=
4
1 −3
−1
3
e 3t
+
4
3 3
1 1
1
0





 +


e−t
y =
4
1
−1
e3t
+
4
3
1
F14 11:40 Final Exam Problem 6
Linear Algebra, Dave Bayer
[6] Find eAt where A is the matrix


1 2 1
A = 0 1 2
0 2 1
eAt =
λ = −1, 1, 3










 +





 +







eAt






0 −1
1
2 −1 −2
0 3 3
t
3t
e−t 
e
e
0
0 2 2
0
2 −2  +
0
0 +
=
4
2
4
0 −2
2
0
0
0
0 2 2
F14 11:40 Final Exam Problem 7
Linear Algebra, Dave Bayer
[7] Solve the differential equation y 0 = Ay where


2 1 2
A =  1 2 1 ,
0 0 1
y =
eAt

1
y(0) =  1 
1










 +





 +












2 2 3
2 −2 −3
0 0
1
t
t
e 
e 
te 
2 2 3 +
−2
2 −3  +
0 0 −1 
=
4
4
2
0 0 0
0
0
4
0 0
0
3t
λ = 3, 1, 1


 




7
−3
1
t
e3t  
et 
te
 −1 
7 +
−3  +
y =
4
4
2
0
4
0
F14 11:40 Final Exam Problem 8
Linear Algebra, Dave Bayer
[8] Express the quadratic form
2x2 + 2xy + 3y2 − 2yz + 2z2
as a sum of squares of orthogonal linear forms.
2
+
2
+
2







1 −1 −1
1
2 −1
2
1
0
1 0 1
1
2
1
1 + 0 0 0 +  2
4 −2 
3 −1  =  −1
A = 1
3
3
−1
1
1
−1 −2
1
0 −1
2
1 0 1

λ = 1, 2, 4
2
1
(x − y − z)2 + (x + z)2 +
(x + 2y − z)2
3
3