F14 8:40 Final Exam Problem 1
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p1
2 2 7 7
Exam 01
Uni
Name
[1] Find the intersection of the following two affine subspaces of R4 .
 
w

1 0 0 0 
 x = 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






 + 







t


F14 8:40 Final Exam Problem 2
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p2
5 6 6 1
Exam 01
[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=
1






F14 8:40 Final Exam Problem 3
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p3
5 6 6 4
Exam 01
[3] Find the inverse of the matrix


1 0 2
A=2 0 1
3 1 2
A−1 =
1








F14 8:40 Final Exam Problem 4
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p4
0 2 8 3
Exam 01
[4] Find An where A is the matrix
A =
3
2
−2 −2
An =
n 


 +
n 



F14 8:40 Final Exam Problem 5
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p5
5 2 8 1
Exam 01
[5] Solve the differential equation y 0 = Ay where
0
3
A =
,
2 −1
y(0) =
y =
1
2





 +


F14 8:40 Final Exam Problem 6
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p6
5 0 4 5
Exam 01
[6] Find eAt where A is the matrix


2 1 2
A = 0 2 1
0 1 2
eAt =










 +





 +







F14 8:40 Final Exam Problem 7
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p7
7 1 3 6
Exam 01
[7] Solve the differential equation y 0 = Ay where


2 0 1
A =  1 1 1 ,
0 1 2
y =


0
y(0) =  1 
1










 +





 +







F14 8:40 Final Exam Problem 8
Linear Algebra, Dave Bayer
[Reserved for Score]
exam01a4p8
2 6 9 8
Exam 01
[8] Express the quadratic form
2x2 − 2xy + 3y2 + 2yz + 2z2
as a sum of squares of orthogonal linear forms.
2
+
2
+
2