126 CHAPTER 4. ROW OPERATIONS

(c)

0 1 0 2 1 2 2

0 3 2 12 1 6 8

0 1 1 5 0 2 3

0 2 1 7 0 3 4



(d)

0 1 0 2 0 1 0

0 3 2 6 0 5 4

0 1 1 2 0 2 2

0 2 1 4 0 3 2



(e)

0 1 0 2 1 1 2

0 3 2 6 1 5 1

0 1 1 2 0 2 1

0 2 1 4 0 3 1

10. Suppose A is an m × n matrix. Explain why the rank of A is always no larger than

min (m,n) .

11. 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.

12. Suppose A is an m × n matrix and that m > n. Show there exists b ∈ Fm such thatthere is no solution to the equation

Ax = b.

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

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

15. Suppose A is an m × n matrix and {w1, · · · ,wk} is a linearly independent set ofvectors in A (Fn) ⊆ Fm. Suppose also that Azi = wi. Show that {z1, · · · , zk} is alsolinearly independent.

16. Show rank (A+B) ≤ rank (A) + rank (B).

17. Suppose A is an m × n matrix, m ≥ n and the columns of A are independent. Sup-pose also that {z1, · · · , zk} is a linearly independent set of vectors in Fn. Show that{Az1, · · · , Azk} is linearly independent.

18. Suppose that A is an m× n matrix 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) .