1. Can a set of vectors that includes the 0 vector be linearly independent?
   a) Yes and I can give an example
   b) Yes but I am unsure of why
   c) No but I am unsure of why not
   d) No and I can explain why not
   
   
   
   
2. Can a matrix span R³?
   a) Yes and I can give an example
   b) Yes but I am unsure of why
   c) No but I am unsure of why not
   d) No and I can explain why not
   
   
   
   
3. The columns of a coefficient matrix span Rⁿ exactly when the augmented matrix reduces to one with a pivot for each column except the equals column
   a) True and I can explain why
   b) True but I am unsure of why
   c) False but I am unsure of why not
   d) False and I can give a counterexample
   
   
   
   
4. To check whether a vector is in the span of other vectors, it suffices to see if they are multiples
   a) True and I can explain why
   b) True but I am unsure of why
   c) False but I am unsure of why not
   d) False and I can give a counterexample
   
   
   
   
5. If a collection of vectors is not l.i. then we could throw away any one vector and still span the same space
   a) True and I can explain why
   b) True but I am unsure of why
   c) False but I am unsure of why not
   d) False and I can give a counterexample
   
   
   
   
6. If vectors are not linearly independent then they can't span R²
   a) True and I can explain why
   b) True but I am unsure of why
   c) False but I am unsure of why not
   d) False and I can give a counterexample
7. Let z be any vector from R³: If we have a given set V of vectors from R³; how many vectors must be in V to guarantee that z can be written as a linear combination of the vectors in V ?
   a) 2
   b) 3
   c) 4
   d) It is not possible to make such a guarantee
   
   
   
   
8. How would we geometrically describe the span of (1,0,0), (0,0,1) and (1,2,3)?
   a) A point
   b) A line segment
   c) A line
   d) R²
   e) R³
   
   
   
   
9. If two vectors are linearly independent, they must be perpendicular (orthogonal).
   (a) True, and I am very confident
   (b) True, but I am not very confident
   (c) False, but I am not very confident
   (d) False, and I am very confident
   
   
   
   
10. Which set of vectors is linearly independent?
    (a) (2, 3), (8, 12)
    (b) (1, 2, 3), (4, 5, 6), (7, 8, 9)
    (c) (-3,1,0), (4, 5, 2), (1, 6, 2)
    (d) None of these sets are linearly independent.
    (e) Exactly two of these sets are linearly independent.
    
    
    
    
11. Lucinda owns two ice cream parlors. The first ice cream shop sells 5 gallons of vanilla ice cream and 8 gallons of chocolate ice cream each day. The daily sales at the second store are 6 gallons of vanilla ice cream and 10 gallons of chocolate ice cream. The daily sales at stores one and two can be represented by the vectors s1 = (5,8) and s2 = (6,10) respectively. In this context, what interpretation can be given to the vector 15s1?
    (a) 15s1 shows the number of people that can be served with 15 gallons of vanilla ice cream.
    (b) 15s1 shows the gallons of vanilla and chocolate ice cream sold by store 1 in 15 days.
    (c) 15s1 gives the total revenue from selling 15 gallons of ice cream at store 1.
    (d) 15s1 represents the number of days it will take to sell 15 gallons of ice cream at store 1.
    
    
    
    
12. The stores are run by different managers, and they are not always able to be open the same number of days in a month. If store 1 is open for c1 days in March, and store 2 is open for c2 days in March, which of the following represents the total sales of each flavor of ice cream between the two stores?
    a) c1s1+c2s2
    b) Matrix([[5,6],[8,10]]).Vector([c1,c2])
    c) Vector([5c1,8c1]) + Vector([6c2,10c2])
    d) All of the above
    e) None of the above
    
    
    
    
13. Lucinda is getting ready to close her ice cream parlors for the winter. She has a total of 39 gallons of vanilla ice cream in her warehouse, and 64 gallons of chocolate ice cream. She would like to distribute the ice cream to the two stores so that it is used up before the stores close for the winter. How much ice cream should she take to each store? The stores may stay open for different number of days, but no store may run out of ice cream before the end of the day on which it closes.
    a) 3 gallons of each kind to store 1 and 4 gallons of each kind to store 2
    b) 3 gallons of vanilla to each store and 4 gallons of chocolate to each store
    c) 15 gallons of vanilla and 24 gallons of chocolate to store 1 and 24 gallons of vanilla and 40 gallons of chocolate to store 2
    d) This cannot be done unless ice cream is thrown out or a store runs out of ice cream before the end of the day
    
    
    
    
14. For full credit, which of the following are true regarding graded problem sets
    a) I must print out all work, including Maple ReducedRowEchelonForm commands and output
    b) I must annotate/explain my methods and reasoning with handwritten comments and/or typed comments.
    c) both a) and b)
    d) neither a) nor b)