 x
1. Give a geometric interpretation for : W= {X : X=  y  , x 2  2 y 2  3z  0 ,x≥0,y≥0}
 
 z 
What do you need to show this is a subspace?
Determine if the above subset is subspace. Show all work.
2. Find the null space for the Matrices:
1
0
C
0

1
3
0 
7

0 1 10 
0 0
1 2
0 1
a. Identify the pivot columns
b. Identify the free variables
c. Find the special solutions for the above matrix.
d. What is the rank of the matrix?
3. For the above matrices A and C,
a. What is the span of the column space?
b. Find the particular solutions to
4
3
Cx=  
0 
 
0 
Describe the set of all solutions to the above
4. Determine if the following are linear Transformations. Find the Kernel and describe the range.
T:
3
→
3
defined by T(x,y,z) = (x + y, –z, x + y +3z)
5. Find an orthogonal basis for
3
1 2 0 
using the column space of D= 0 5 3 


1 0 2 
Make an orthonormal basis from the columns of D.
Find the vector [ 1 2 3]T as a linear combination of the orthonormal basis above.
TRUE FALSE QUESTIONS:
6.
2 0 
2 0 
1


1 2
5 2
A  1  2, B   , C  0  2 D  
,E  


3
3 4
2 0
3 2 
1 2 
True or False
||AB|| =110
True or False
(DE+E)B = (ED)B+DB
True or False
CTCis symetric
7.
Which of the functions below is a Linear Transformation?
True or False
x   x

F   1     2 1 2 
  x2    x2  x2 
True or False
  x   x 
F   1     1 
  x2    1 
True or False
 x  x12 
  x1    1

F      2 x 2  2 x1 
  x2   

0

