linearly independent vectors question

Question

Given three vectors x₁ = [2 4 8]^T, x₂ = [1 -11]^T, x₃ = [1 1 4]^T, test if they are linearly independent. Write x₁ as a linear sum of x₂ and x₃

Answer 1

To test for linear independence, evaluate the determinant of a matrix whose rows (or columns) are the 3 given vectors.  If the determinant is zero, the vectors are linearly dependent.  If the determinant is not zero, the vectors are linearly independent, and you cannot express any of them as a linear combination of the other two.  I obtained a nonzero value for the determinant, so it is not possible to obtain x₁ as a linear combination of x₂ and x₃.