aGive an example of a 3 3 matrix whose column space is a pl

(a)Give an example of a 3 × 3 matrix whose column space is a plane through the origin in R3 .

(b) What kind of geometric object is the nullspace of your matrix? Explain your answer.

(c) What kind of geometric object is the row space of your matrix? Explain your answer.

Solution

a)- Any 3*3 matrix that has two linearly independent columns will have a column space that is a plane

passing through the origin.

(b) If a 3*3 matrix has a column space that is a plane passing through the origin, then the dimension of

the null space will be equal to one. Therefore, the nullspace will be a line passing through the origin.

(c) The row space will be one-dimensional and, therefore, will be a line passing through the origin.