Problem: find a $1\times3$ matrix whose nullspace consists of all vectors in $\mathbf{R}^3$ such that $x_1 + 2\times x_2+4\times x_3 = 0$. Answer from book (Linear Algebra and Applications): $(1,2,4)$
How can nullspace contain all vectors in $\mathbf{R}^3$ if it should contain only 2 special solution at most(number of columns - rank)?
It says all vectors such that $x_1 + 2x_2+4x_3 = 0$, which is a $2$-dimensional subspace of $\mathbf{R}^3$.
For example you can pick the matrix $\begin{bmatrix}1 & 2 & 4\end{bmatrix}$.