Decide with proof whether linearly independent over F

Question

Consider the following vectors w1=(1,1,1,1) w2 = (1,0,1,0) w3 = (1,1,2,2) w4 = (-1,1,4,1) Decide with proof whether the 4 vectors are linearly independent. I am unsure of how to do this

Answer 1

To decide if the four vectors $(1,1,1,1)^T, (1,0,1,0)^T,(1,1,2,2)^T,(-1,1,4,1)^T$, you need to show that $$c_1\pmatrix{1\\1\\1\\1} + c_2\pmatrix{1\\0\\1\\0} + c_3\pmatrix{1\\1\\2\\2} + c_4\pmatrix{-1\\1\\4\\1} = \pmatrix{0\\0\\0\\0}$$ only has the trivial solution $c_1 = c_2 = c_3 = c_4 = 0$. The left-side can be rewritten to yield $$\pmatrix{c_1 + c_2 + c_3 - c_4\\ c_1 + c_3 + c_4\\c_1 + c_2 + 2c_3 + 4c_4\\ c_1 + 2c_3 + c_4} = \pmatrix{0\\0\\0\\0},$$ which we can write as a linear system $$\pmatrix{1 & 1 & 1 & -1 & 0\\ 1 & 0 & 1 & 1 & 0\\1 & 1 & 2 & 4 & 0\\1 & 0 & 2 & 1 & 0},$$ from which you can use elementary row operations to find the reduced-row echelon form to determine the solution.