7.4. EXERCISES 211

(b) A is a 3× 4 matrix, rank (A) = 3 and rank (A|b) = 2.

(c) A is a 4 × 2 matrix, rank (A) = 4 and rank (A|b) = 4. Hint: This says b is inthe span of the columns and the columns must be independent.

(d) A is a 5 × 5 matrix, rank (A) = 4 and rank (A|b) = 5. Hint: This says b is notin the span of the columns.

(e) A is a 4× 2 matrix, rank (A) = 2 and rank (A|b) = 2.

29. Suppose A is an m×n matrix in which m ≤ n. Suppose also that the rank of A equalsm. Show that A maps Fn onto Fm. Hint: The vectors e1, · · · , em occur as columnsin the row reduced echelon form for A.

30. Suppose A is an m×n matrix in which m ≥ n. Suppose also that the rank of A equalsn. Show that A is one to one. Hint: If not, there exists a vector, x such that Ax = 0,and this implies at least one column of A is a linear combination of the others. Showthat this would require the column rank to be less than n.

31. Explain why an n× n matrix A is both one to one and onto if and only if its rank isn.

32. If you have not done this already, here it is again. It is a very important result ofSylvester. Even if you have done it, a review is a good idea. Suppose A is an m × nmatrix and B is an n× p matrix. Show that

dim (ker (AB)) ≤ dim (ker (A)) + dim (ker (B)) .

Hint: Consider the subspace, B (Fp) ∩ ker (A) and suppose a basis for this subspaceis {w1, · · · ,wk} . Now suppose {u1, · · · ,ur} is a basis for ker (B) . Let {z1, · · · , zk}be such that Bzi = wi and argue that

ker (AB) ⊆ span (u1, · · · ,ur, z1, · · · , zk) .

Here is how you do this. Suppose ABx = 0. Then Bx ∈ ker (A) ∩ B (Fp) and so

Bx =∑k

i=1Bzi showing that

x−k∑

i=1

zi ∈ ker (B) .

33. Recall that every positive integer can be factored into a product of primes in a uniqueway. Show there must be infinitely many primes. Hint: Show that if you have anyfinite set of primes and you multiply them and then add 1, the result cannot bedivisible by any of the primes in your finite set. This idea in the hint is due to Euclidwho lived about 300 B.C.

34. There are lots of fields. This will give an example of a finite field. Let Z denote the setof integers. Thus Z = {· · · ,−3,−2,−1, 0, 1, 2, 3, · · · }. Also let p be a prime number.We will say that two integers, a, b are equivalent and write a ∼ b if a − b is divisibleby p. Thus they are equivalent if a − b = px for some integer x. First show thata ∼ a. Next show that if a ∼ b then b ∼ a. Finally show that if a ∼ b and b ∼ cthen a ∼ c. For a an integer, denote by [a] the set of all integers which is equivalentto a, the equivalence class of a. Show first that is suffices to consider only [a] fora = 0, 1, 2, · · · , p− 1 and that for 0 ≤ a < b ≤ p− 1, [a] ̸= [b]. That is, [a] = [r] wherer ∈ {0, 1, 2, · · · , p− 1}. Thus there are exactly p of these equivalence classes. Hint: