How to write linearly dependent column in a matrix in terms of linearly independent columns?

I have a large mxn matrix, and I have identified the linearly dependent columns. However, I want to know if there's a way in R to write the linearly dependent columns in terms of the linearly independent ones. Since it's a large matrix, it's not possible to do based on inspection.

Here's a toy example of the type of matrix I have.

> mat <- matrix(c(1,1,0,1,0,1,1,0,0,1,1,0,1,1,0,1,0,1,0,1), byrow=TRUE, ncol=5, nrow=4)
> mat
      [,1] [,2] [,3] [,4] [,5]
[1,]    1    1    0    1    0
[2,]    1    1    0    0    1
[3,]    1    0    1    1    0
[4,]    1    0    1    0    1

Here it's obvious that x3 = x1-x2, x5=x1-x4. I want to know if there's an automated way to get that for a larger matrix.

Thanks!