### Rank and Nullity

1. How do you find the rank of a matrix?
2. 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.

3. 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

4. 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
5. The standard matrix for T(x,y)=(-x,y) is:
6. The linear transformation T(x; y) = (x + 2y, x - 2y), can be written as a matrix transformation
7. Which of the following is not a linear transformation?
a) T(x,y) = (x,y+1)
b) T(x,y)=(x-2y,x)
c) T(x,y)=(4y,x-2y)
d) T(x,y)=(x,0)
e) All are linear transformations
8. What is the range of T(v)=A(v) where A=matrix([[1,2,3],[2,-2,0]])
a) All of R3
b) All of R2
c) A line in R2
d) A line in R3
e) A plane in R3
9. What is the kernal of T(v)=A(v) where A=matrix([[1,2,3],[2,-2,0]])
a) All of R3
b) A point in R 3
c) A line in R2
d) A line in R3
e) A plane in R3
10. 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 x-axis.
(c) Aw is a reflection of w in the y-axis.
(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
11. Do the column vectors in this table span R2? Do they form a basis for R2?
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
12. 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.
13. If the store continues to stock all three of these blends, which special-request 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).