What are the values of parameters $s, t \in \mathbb{R}$ such that
$$c_1 = (5, 7, s, 2), \quad c_2 = (1, 3, 2, 1), \quad c_3 = (2, 2, 4, t)$$
of $\mathbb{R}^4$ are linearly independent?
There are a few things that I lack clarity on. First of all, am I supposed to write a $4\times 3$ matrix or a $3 \times 4$?
Initially, I wrote a $3 \times 4$ matrix (3 rows) but I couldn't reduce the matrix enough to solve for the 2 variables. I would be able to solve with 1 unknown but with 2 I get both of them on the same row.
When I wrote it as a $4 \times 3$ (4 rows) matrix I solved it by row-echelon reduction and got $t$ on the bottom row by itself and $s$ with another value and I managed to show that for $$ t\neq\frac{1}{2}, s\neq0 $$ the vectors are linearly independent. But doesn't that spawn 4 equations in a 3D space instead of 4D?
I was also told that I can create a submatrix and check if the submatrix is linearly independent (or not) by calculating its determinant and then do this 3 times in this case and show that it is linearly independent. However, this is not a square matrix so I am not sure if determinant operations hold.