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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.

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

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