Linear Algebra (Symmetric Matrix)

[MoBaT]

A 3x3 symmetric matrix has a null space of dimension one containing the vector (1,1,1). Find the bases and dimensions of the column space, row space, and left null space.

I understand how to get the Dim of Col(A), Row(A), and Nul(A^T) but how do i get the bases with just knowing the dimension of one vector? How should I approach this?

 

[chiro]

Hey MoBaT.

Can you pick any linearly independent basis?

 

[MoBaT]

Hey MoBaT.

Can you pick any linearly independent basis?

Dosen't say anything against it. I know what the answer is becuase he gave it to us. It was like:

Dim of Col(A) = 2, dim Row(A) = 2, Dim Nul(A^T) = 1
Basis Row(A) = {[-1 1 0], [-1 - 1]}. Because A is symmetric, Col(A) = Row(A) and Nul(A^T) = Nul(A).

I understand everything after the Basis row(A) but do not understand how he got that Row(A)

 

[MoBaT]

Found out how to do it. Pretty much fill in the rest of the information with the identity matrix.