show that the zero matrix is the only 3x3 matrix whose null space has dimension 3

Question

anonymous · Answer

To restate the problem: You have a 3 x 3 matrix $A$, where the nullspace of $A$ (i.e., the set of all vectors $x$ for which $Ax = 0$) has the dimension 3. You want to prove that $A=0$.

Note that in the system $Ax = 0$ the vector $x$ has to have 3 elements (because otherwise it couldn't be multiplied on the left by a matrix with 3 columns). So a vector $x$ in the nullspace is a vector in a 3-dimensional vector space, for example $\mathbb{R}^3$. And per the statement of the problem the nullspace of $A$ (the set of all such $x$) also has 3 dimensions.

From there you can go several different ways to find the solution:

First, you can figure out how the 3-dimensional nullspace of $A$ relates to the overall 3-dimensional vector space in which it's contained. Then from the fact that $Ax = 0$ you can figure out what $A$ must be.

Second, you can remember the relationship between the dimension of the column space of a matrix (which is the rank $r$ of the matrix) and the dimension of the nullspace of a matrix, and how they both relate to the dimension of the overall vector space that contains the nullspace and column space. If you know the dimension of the nullspace of $A$ (in this case, 3) and the dimension of the vector space (also 3) then you know the dimension of the column space $A$. And knowing the dimension of the column space tells you what kind of columns are in the matrix $A$, and hence what kind of matrix $A$ is.

Third, you can remember that the column space and the nullspace of a matrix are orthogonal subspaces: every vector in the nullspace is orthogonal (perpendicular) to every vector in the column space, and vice versa. Then, knowing that the dimension of the nullspace is 3 and how it relates to the overall 3-dimensional vector space that contains it, you can figure out what the nullspace is and then figure out what vectors would be orthogonal to it. That gives you the column space, and from that you can figure out what the matrix $A$ looks like.

I hope I didn't overwhelm you with hints; I think it's useful to know that there's multiple paths to a solution, and they all lead to the same place. If you need anything else to solve this, let me know.

anonymous · Answer

Simply notice that the dimensions of the collumn space and the nullspace of a 3x3 matrix must sum 3, so if the dimension of the nullspace is 3 then there are no independent vectors in the collumns of the matrix. In other words, the rank of the matrix is 0, and the only matrix which satisfies this property is the zero matrix.