18.6. EXERCISES 407

(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 in the

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 not in

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

41. 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 columns inthe row reduced echelon form for A.

42. 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. Showthis would require the column rank to be less than n.

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

44. For M a matrix, ker(M) consists of all vectors x such that Mx= 0. Suppose A is anm×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) .

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

ki=1 Bzi showing that x−∑

ki=1zi ∈ ker(B) .

45. Explain why Ax= 0 always has a solution even when A−1 does not exist.

(a) What can you conclude about A if the solution is unique?(b) What can you conclude about A if the solution is not unique?

46. Let A be an n×n matrix and let x be a nonzero vector such that Ax= λx for somescalar λ . When this occurs, the vector x is called an eigenvector and the scalar λ

is called an eigenvalue. It turns out that not every number is an eigenvalue. Onlycertain ones are. Why? Hint: Show that if Ax= λx, then (A−λ I)x= 0. Explainwhy this shows that (A−λ I) is not one to one and not onto.

47. Let A be an n×n matrix and consider the matrices{

I,A,A2, · · · ,An2}. Explain why

there exist scalars, ci not all zero such that ∑n2

i=1 ciAi = 0. Then argue there exists apolynomial, p(λ ) of the form

λm +dm−1λ

m−1 + · · ·+d1λ +d0

such that p(A) = 0 and if q(λ ) is another polynomial such that q(A) = 0, then q(λ )is of the form p(λ ) l (λ ) for some polynomial, l (λ ) . This extra special polynomial,p(λ ) is called the minimal polynomial. Hint: You might consider an n×n matrixas a vector in Fn2

. What would be a basis for this set of matrices?