F14 8:40 Final Exam Problem 1
Linear Algebra, Dave Bayer
[1] Find the intersection of the following two affine subspaces of R4 .
 
w

x
1 0 0 0 
  = 1
0 0 0 1  y
1
z






w
1
1 1  x




  =  2 + 1 0 r
 y
 −2 
0 1 s
z
1
1 1


w


 x
  = 

 y


z


 


w
1
0
 x
 


  =  0  +  1 t
 y
0
 −1 
z
1
0










 + 







t


F14 8:40 Final Exam Problem 2
Linear Algebra, Dave Bayer
[2] Find the 3 × 3 matrix A that maps the vector (1, 2, 1) to (3, 6, 3), and maps each point on the plane
x + y + z = 0 to the zero vector.


A=


3 3 3
1
A = 6 6 6
4
3 3 3
1






F14 8:40 Final Exam Problem 3
Linear Algebra, Dave Bayer
[3] Find the inverse of the matrix


1 0 2
A=2 0 1
3 1 2
A−1 =


−1
2 0
1
−1 −4 3 
A =
3
2 −1 0
1








F14 8:40 Final Exam Problem 4
Linear Algebra, Dave Bayer
[4] Find An where A is the matrix
A =
3
2
−2 −2
n 
An =
λ = −1, 2
A
n
(−1)n
=
3

 +

−1 −2
2
4
2n
+
3
4
2
−2 −1
n 



F14 8:40 Final Exam Problem 5
Linear Algebra, Dave Bayer
[5] Solve the differential equation y 0 = Ay where
0
3
A =
,
2 −1
y(0) =
y =
λ = −3, 2
e
At
e−3t
=
5
2 −3
−2
3
e 2t
+
5
3 3
2 2
1
2





 +


e−3t
y =
5
−4
4
e 2t
+
5
9
6
F14 8:40 Final Exam Problem 6
Linear Algebra, Dave Bayer
[6] Find eAt where A is the matrix


2 1 2
A = 0 2 1
0 1 2
eAt =
λ = 1, 2, 3










 +





 +







eAt






0
1 −1
0 3 3
1 −2 −1
3t
et 
e
0 1 1
0
1 −1  + e2t  0
0
0 +
=
2
2
0
0
0
0 −1
1
0 1 1
F14 8:40 Final Exam Problem 7
Linear Algebra, Dave Bayer
[7] Solve the differential equation y 0 = Ay where


2 0 1
A =  1 1 1 ,
0 1 2
y =
eAt

0
y(0) =  1 
1










 +





 +












1 1 2
3 −1 −2
1 −1 0
t
t
e 
e 
te 
1 1 2 +
−1
3 −2  +
1 −1 0 
=
4
4
2
1 1 2
−1 −1
2
−1
1 0
3t
λ = 3, 1, 1


 




3
−3
−1
t
e3t  
et 
te
 −1 
3 +
1 +
y =
4
4
2
3
1
1
F14 8:40 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