Give an example of a 3 x 3 matrix whose column space is a pl

Give an example of a 3 x 3 matrix whose column space is a plane through the origin in R^3. What kind of geometric object is the nullspace of your matrix? Explain your answer. What kind of geometric object is the row space of your matrix? Explain your answer.

Solution

Since no equations are given , I am assuming that this is more of a conceptual question rather than a matrix algebra question that can be solved

a) If a matrix 3x3 has 2 linearly independent columns then the matrix will have a column space which is a plane passing through region , so if the vector x1 x2 x3 is non zero then theya re linear dependent if and only if x1=x2=x3=0

b)The dimension of the null space of a plane passing through the origin will be 1 , so the nullspace of the 3x3 matrix would be a line through the origin as the standard definiton of the null space of am matrix is the set of all vectors x for which Mx = 0

c) the row space would again be a line passing through the origin as the row space will be one dimensional in this case . The row space of R3 would be given by [r1,r2,r3]