Is it true to say that any matrix in row echelon form, excluding the zero matrix, with columns as vectors, has linearly independent vectors (of the original matrix before performing elementary row operations)? I'm wondering if I can use this as a faster method of checking for linear independence between the vectors of a set.
Thanks! 🙂
Best Answer
If you want to know whether all column vectors are linearly independent, then no, $$\begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$$
is an example of a matrix in row-echelon form that has rank 2, which means it not all 3 column vectors can be linearly independent.
You can put any matrix in row echelon form, and then it is easier to determine the rank (number of linearly independent column vectors) of a matrix: The rank stays the same under Gaussian elimination, so a matrix has rank $n$ if and only if its row echelon form has rank $n$.
If you want to know whether a matrix has at least one linearly independent column vector, then any nonzero matrix has at least rank 1, as @DonAntonio said in the comments.