null space of a matrix A I am studying about robotics grasping and I came across null space, which I am not clear about. $$$$


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*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?

 A: 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.
A: The null space always exists. It may be reduced to $0$ or not. If it is reduced to $0$, we have only trivial solutions.
As an $(n\times n)$ matrix is the matrix of a linear map from $\mathbf R^n$ into itself (an endomorphism of $\mathbf R^n$), saying the null space of $A$ is $0$ means the linear map is injective (hence bijective). Incicentally, the null space, in terms of linear maps, is usually known as the kernel of the linear map.
