Can a matrix whose rows are linear combinations of rows of a full-rank matrix be guaranteed to be full rank?

Question

Let's say I have a matrix $M$, whose rows are linearly independent and has full rank, that can be described as:

$$M=\begin{bmatrix} m_1 \\ m_2 \\ ... \\ m_n \\ \end{bmatrix}$$

Now, let's say I create a matrix $N$, whose rows are linear combinations of the exactly two rows in $M$, such that:

$$N=\begin{bmatrix} c_1m_1 + c_2m_2 \\ c_3m_1 + c_4m_3 \\ ... \\ c_{x-1}m_{n-1} + c_{x}m_n \\ \end{bmatrix}$$

Here, $c_1...c_x$ are non-zero constants that we can choose. Does there exist a set of values for these constants such that $N$'s rows are linearly independent?

Answer 1

Let $n=4$. Then $M$ has four rows, and they generate a vector space of dimension 4. $N$ has six rows, and they all live in that 4-dimensional vector space, so they can't be linearly independent.

This argument works for all $n\ge4$.