Let G be a matrix with only nonnegative elements with linearly independent columns. Then there exists a column, g of G such that the projection of g on the remaining columns of G is a positive linear combination of those columns.

Is the above a correct statement?

Sorry my question was not clearly stated. I will ask it more clearly.

Let $G$ be a matrix with only nonnegative elements with linearly independent columns. Then there exists a column, ${\bf g}$ of $G$ such that the orthogonal projection of ${\bf g}$ on the remaining columns of G is a nonnegative linear combination of those columns. In other words, for a suitable column ${\bf g}$ of $G$ the vector ${\bf x}$ that minimizes $\parallel G^* {\bf x} - {\bf g}\parallel$ has nonnegative elements, where $G^*$ is a matrix consisting of all the columns of $G$ except ${\bf g}$.

Is the above a correct statement?