1. State the negations of the following propositions:
a) (Background: A is an m n matrix and b2Rm.) There exists an x2Rn such that Ax = b.
b) (Background: f is a function on R.) For all x;y2R, if x6= y, then f(x)6= f(y).
c) (Background: G is a set of some n n matrices and contains the identity matrix I.) For every A2G,
there exists a B2G such that AB = BA = I.
d) (Background: f is a function on R and x2R.) For every > 0, there exists a > 0 such that for every
x0 2R, if jx x0j< , then jf(x) f(x0)j< .
2. Determine whether each of the following subsets of R3 is a vector subspace of R3.
a) f(0;0;0)g.
b) f(x;y;z)2R3 : x+ 2y + 3z = 1g.
c) f(x;y;z)2R3 : x+ 2y + 3z = 0 and 3y + 4z = 0g.
d) f(x;y;z)2R3 : x2 +y2 +z2 1g.
e) f(t;2t;5t) : t2Rg.
3. Show that v1 = (1;3) and v2 = (3;5) form. a basis of R2 and represent (1;1) and (0;6) as linear
combinations of v1 and v2.
4. Determine the rank and dimension of kernel of each of the following matrices:
5. (ECON 8013 only) Show that if V is a vector subspace of Rn, then the zero vector is in V.
6. (ECON 8013 only) There are 52 playing cards; let S be the set of all of them. A \deck" is a particular
order (or permutation) of the 52 cards. Mathematically, a deck can be represented by (c1;c2;:::;c52) where
c1;c2;:::;c52 are all the elements of S. The interpretation is that c1 is the rst card in the deck, c2 is the
second card, and so on. Let be the set of all possible decks.
a) Is a subset of S52? Is S52 a subset of ?
b) A deck of cards is distributed to four players (Adam, Bob, Chris and David), each receiving 13. The set
of all cards that a player receives is called his hand. Let A be the set of possible hands of Adam. Is A a
subset of ? Is A a subset of S52? Is A a subset of S13?
c) Here is how the cards are distributed. Players take cards in the cycle Adam-Bob-Chris-David-Adam-...
Each player takes one card when it is his turn. (For example, Adam takes the rst card, Bob takes the
second, Chris takes the third, David takes the fourth, and then Adam takes the fth.) Given a deck
(c1;c2;:::;c52), what is Adam’s hand?
d) What is the subset of corresponding to the event that Adam has }A (Ace of diamond)?
e) What is the subset of corresponding to the event that Adam has both }A and }K?