Rank and Nullity
 How do you find the rank of a matrix?

Suppose a 4 4 matrix A has rank 3. How many solutions does the system Ax = b
have?
(a) 0
(b) 1
(c) Infinite
(d) Not enough information is given.
 Let A=Matrix([[1,0,2,1],[0,1,3,1],[2,1,1,1]]).
What is the dimension of the nullspace of A?
a) 0
b) 1
c) 2
d) 3
e) 4
f) Infinite
Transformations
 Define T(v)=A(v) where A=Matrix([[1,0],[0,1]]). Then T(v)
a) reflects v about the y axis
b) reflects v about the x axis
c) rotates v clockwise 90 degress
d) rotates v counterclockwise 90 degrees
e) None of the above
 The standard matrix for T(x,y)=(x,y) is:
 The linear transformation T(x; y) = (x + 2y, x  2y),
can be written as a matrix
transformation
 Which of the following is not a linear transformation?
a) T(x,y) = (x,y+1)
b) T(x,y)=(x2y,x)
c) T(x,y)=(4y,x2y)
d) T(x,y)=(x,0)
e) All are linear transformations
 What is the range of T(v)=A(v) where A=matrix([[1,2,3],[2,2,0]])
a) All of R^{3}
b) All of R^{2}
c) A line in R^{2}
d) A line in R^{3}
e) A plane in R^{3}
 What is the kernal of T(v)=A(v) where A=matrix([[1,2,3],[2,2,0]])
a) All of R^{3}
b) A point in R ^{3}
c) A line in R^{2}
d) A line in R^{3}
e) A plane in R^{3}

When we map w to Aw and w is an eigenvector of A, what is the geometric effect?
(a) Aw is a rotation of w.
(b) Aw is a reflection of w in the xaxis.
(c) Aw is a reflection of w in the yaxis.
(d) Aw is parallel to w but may have a different length.
Mixing
Howard's store sells three blends of
our: standard, extra wheat, and extra soy. Each
is a blend of whole wheat
flour and soy
flour, and below shows how many
pounds of each type of
flour is needed to make one pound of each blend.
Standard Blend 
Extra Wheat 
Extra Soy 

.5 
.8 
.3 
whole wheat flour 
.5 
.2 
.7 
soy flour 
 Do the column vectors in this table span R^{2}? Do they form
a basis for R^{2}?
a) Yes they span and form a basis
b) They span but do not form a basis
c) They do not span but they do form a basis
d) They do not span nor do they form a basis
 To save rent money, the store will be moving to a
smaller space and will need to cut
back on inventory. If possible, the manager would like to only stock two of
these
blends, and make the third from those as necessary. Which blends can be made
from
the others?
(a) Standard Blend can be made from Extra Wheat Blend and Extra Soy Blend.
(b) Extra Wheat Blend can be made from Standard Blend and Extra Soy Blend.
(c) Extra Soy Blend can be made from Standard Blend and Extra Wheat Blend.
(d) Any one blend can be made from the other two.
 If the store continues to stock all three of these blends, which specialrequest blends
could be made from these three?
(a)Any special request could be accomodated mathematically
by mixing the right combination of
these three blends.
(b) It would be possible to make any blend that is between 30% and 80% whole wheat.
(c) It would be possible to make a broader range of blends than what is described in
answer (b), but there are still some blends that would not be possible.
(d) It would be possible to satisfy some special requests, but not all of
the ones described in answer (b).