1. Let Then A and B are row equivalent (You don't have to check this). For the following problems
a, b, c, d, and e, justify your answers. Find rank A and dim Nul A. Find a basis for Col A. Find a
basis for Col A. Find a basis for Row A Find a basis for Nul A.
Solution
(a) A and B are row equivalent.so their ranks are same.
Thus rank(A) = rank(B) = 3 (first second and fifth columns of B are independent and other
columns are in the span of these vectors)
dim (nullA) = no of columns - rank(A) = 6 - 3 = 3
(b) a basis of column space of A are first second and fifth columns of A (since A and B are row
equivalent and B has first second and fifth columns as basis of its column space)
(c)basis of row space of A must contain three vectors(since rank = 3)
Easy to see first and second rows are independent and third row is in their span(R3 = R1-R2)
Hence R1,R2,R4 is a basis
(d)A basis of null space of B is (3,0,0,0,0,-1) , (0,1,0,-1,0,0) , (0,0,1,0,0,1).
Since these are in null space of B,are independent,and dimension of nullspace of B is 3
Now A and B are row equivalent So a basis of A is
(3,0,0,0,0,-1) , (0,1,0,-1,0,0) , (0,0,1,0,0,1)