This question is on null space column space row space and nu

This question is on null space, column space, row space, and nullity of a matrix.

Suppose A is a 3 times 3 matrix whose null space is a line through the origin in 3-space. Can the row or column space of A also be a line through the origin? Explain why or why not.

Solution

Since the null space is a line through the origin in 3-space, So the nullity of the matrix is 2, and thus rank is 1.

So there is only one linearly independent row (column) in A.

And hence the row (column) space of A is a line through the origin