I'm learning Linear Algebra using Gilbert Strang's lectures and at lecture 14 he said the following: "Imagine two perpendicular lines in R3. Can they be the row space and the nullspace? No.". So the answer is no but I don't understand why. Say we put 2 such vectors in a 3x2 matrix. The row space is a line, so the dimension of the row space is 1, so the rank is 1. The nullspace is also a line, so n - rank = 1. Therefore n is 2, which is right given the matrix is 3x2. I think my logic is flawed, since my answer is wrong. Can someone explain? Maybe I have to use the fact that the row space and nullspace are orthogonal?

  • $\begingroup$ The sum of the dimensions of the nullspace and the column space is the dimension of your vector space. If one of them is one the other one has to be two.The column space is the range of your transformation. $\endgroup$ – GReyes Feb 5 '19 at 21:53
  • $\begingroup$ @GReyes That should really be an answer instead of a comment. $\endgroup$ – amd Feb 5 '19 at 22:09
  • $\begingroup$ The problem with your logic is that your rowspace and nullspace are actually lines in $R^2$, not $R^3$ (think about what kinds of $x$'s can be multiplied by a 3x2 matrix). $\endgroup$ – user474330 Feb 6 '19 at 6:59

This follows from the Rank-Nullity theorem. But specific to the example you reasoned in your question, when you formulated your 3 x 2 matrix (assuming you mean rows x columns), the Row space will be the vector sub-space spanned by the rows of your matrix viewed as vectors. The nullspace will be solutions of the equation $$Ax=0$$ Thus, your rows will be either 2 linearly independent vectors and 1 linearly dependent vector meaning that your row space has dimension 2 and your nullspace has dimension zero. Or, your rows will be 1 linearly independent vector (assuming you haven't chosen the zero matrix) and 2 linearly dependent vectors (multiples of that first row-vector), in which case your row space has dimension 1 and your nullspace has dimension 1. Keep in mind that in this example your vectors live in $\mathbb{R}^2$. Something similar happens when you consider and a 2 x 3 matrix whose row vectors live in $\mathbb{R}^3$. Can you produce an example matrix of this form and try to work it out?


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