# How to find basis and dimensions of ColA, RowA and NulA^T from NulA?

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

I know that, since the matrix is 3x3 and the dimension of the null space is one, the dimension of the column space would be 2. I also know that since the matrix is symmetric, the left null space would be the same as the column space. But I'm having trouble figuring out how to find a basis without having the actual matrix.

• It's the dimension of the column space that would be equal to $2$, from the rank-nullity theorem. – Bernard May 6 at 18:55
• Hint: The null space of any matrix is the orthogonal complement of its row space, and then use symmetry. – amd May 6 at 19:09