Suppose that A 1 2 3 4 2 2 1 1 3 4 1 0 1 2 2 0 3 3 3 0 and

Suppose that A = [1 2 3 4 2 2 -1 1 3 4 -1 0 -1 -2 -2 0 3 3 3 0] and B = [1 0 1 2 2 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0] are row equivalent. What is the dimension of the column space of A? What is the dimension of the row space of A? What is the dimension of the null space of A? What is the rank of A? Give a basis for the column space of A. Give a basis for the row space of A. Give a basis for the null space of A.

Solution

a.

Dimension of column space of A =dimension of column space of B=2

Because they are rwo equivalent

b.

Dimension of row space of A=dimension of column space of A=2

c.

BY rank nullity theorem

rank(A)+nullity=5

rank(A)=2

2+nullity=5

nullity=3

Hence, dimension of null space of A=3

d.

rank(A)=dimension of row space of A=2

e.

Basis for column space of A=first two rows of A

f. Basis of row space of A=first two rows of A

g.

Ax=0 and Bx=0 have teh same set of solutions

Let, x=[p q r s t]

Bx=0 gives

p+r+2s+2t=0 ie p=-r-2s-2t

q+r+s=0    ie q=-r-s

So,

x=(-r-2s-2t,-r-s,r,s,t)=r(-1,-1,1,0,0)+s(-2,-1,0,1,0)+t(-2,0,0,0,1)

HEnce basis fo null space of A

={(-1,-1,1,0,0),(-2,-1,0,1,0),(-2,0,0,0,1)}