I am studying about robotics grasping and I came across null space, which I am not clear about. $$$$
- The null space of a matrix, $A$, $N(A)$ is the vector $x$ such that $A⋅x=0$. If $x$ is zero vector only, then the solution is said to be trivial, if solutions other than zero exists, they are called non-trivial solutions right? $$$$
- What does $N(A) \ne 0$ mean? Does it mean that there are non-trivial solutions also? $$$$
- What does it mean when we say there exists $N(A)$? Does it mean it we have trivial solutions or non-trivial solutions?
Best Answer
1.) Yes, but $N(A)$ is a vector space (a set of vectors with vector space structure, not just a vector unless $N(A)=\{\vec{0}\}$).
2.) If $N(A)\neq 0$, it means you have non-trivial solutions to $Ax=0$ which is the same as saying $x\to Ax$ is not one-to-one
3.) $N(A)$ is always non-empty: there is always at least one vector (the zero vector) that maps to $\vec{0}$. This is why the zero vector is called a trivial solution. There are a lot of equivalent ways of saying $Ax=0$ has trivial/non-trivial solutions. For example, $Ax=0$ having only trivial solution (only $x=\vec{0}$ satisfies it) if and only if $A$ is invertible.