Suppose that A is a 3 times 3 matrix whose nullspace is a line through the origin in R3. Can the row space of A also be a line through the origin? How about the column space of A? If yes, give an example, if no, explain why. Solution Let A be a n-by-m matrix. Then 1. rank(A) = dim(row space(A)) = dim(col space(A)),.....................(1) also; we have rank-nullity theorem as following..... rank(A) + nullity(A) = number of column in A .....................................................(2) {nullity is the dimension of null space of A} now; given that null space is line . and ; a line has dimension = 1. SO; nullity is = 1.    ........(3) given matrix A has 3 column.        .....................................................(4) using conditions (3) and (4) in equation (3) ae get ; rank(A) + 1 =   3 rank (A) = 2. now notinc ethe equattion (1).we have rank(A) = dim (row space (A) ) . so;                       dimension(row space (A) ) = 2. ................(5) so; row space can\'t be line passing though the origin. because , if row space had been a line then it\'s dimension would have been =1 . but ; we have justs dimension is = 2. now ; we observe the equation (1) and equation (5) ; we get; dim(row space(A)) = dim(col space(A)) = 2. .

1. Suppose that A is a 3 times 3 matrix whose nullspace is a line through the origin in R3. Can the
row space of A also be a line through the origin? How about the column space of A? If yes, give
an example, if no, explain why.
Solution
Let A be a n-by-m matrix. Then
1. rank(A) = dim(row space(A)) = dim(col space(A)),.....................(1)
also; we have rank-nullity theorem as following.....
rank(A) + nullity(A) = number of column in A .....................................................(2)
{nullity is the dimension of null space of A}
now; given that null space is line . and ; a line has dimension = 1. SO; nullity is = 1. ........(3)
given matrix A has 3 column. .....................................................(4)
using conditions (3) and (4) in equation (3) ae get ;
rank(A) + 1 = 3
rank (A) = 2. now notinc ethe equattion (1).we have rank(A) = dim (row space (A) ) .
so; dimension(row space (A) ) = 2. ................(5)
so; row space can't be line passing though the origin. because , if row space had been a line then
it's dimension would have been =1 . but ; we have justs dimension is = 2.
now ;
we observe the equation (1) and equation (5) ; we get;
dim(row space(A)) = dim(col space(A)) = 2.