Suppose that $\boldsymbol{A}$ is a $m\times n$ matrix, $\boldsymbol{x}$ is a $n\times 1$ vector. we have $$\boldsymbol{Ax} = 0.$$
We denote the first $n_1$ elements of $\boldsymbol{x}$ as $\boldsymbol{x}_1$, and the remaining part as $\boldsymbol{x}_2$. For the the partitions $\boldsymbol{x}^{T} = (\boldsymbol{x}_1^{T}, \boldsymbol{x}_2^{T})$, $\boldsymbol{A}=(\boldsymbol{A_1}, \boldsymbol{A_2}),$ $$\boldsymbol{Ax = A_1x_1+A_2x_2=0}$$
If n>m, we can not conclude $\boldsymbol{x}=0$. We expect a conclusion for $\boldsymbol{x}_1=0$.
I am trying to figure out this question: If $n_1<m$, under what conditions on A, we can conclude the first part $\boldsymbol{x}_1=0$?