Are the vectors \[ \alpha_1 = (1, 1, 2, 4), \quad \alpha_2 = (2, -1, -5, 2), \] \[ \alpha_3 = (1, -1, -4, 0), \quad \alpha_4 = (2, 1, 1, 6) \] linearly independent in \(\RR^4\)?
Solution
They are linearly dependent. Indeed, \[ \frac13\alpha_1-\frac23\alpha_2+\alpha_3=0. \] Equivalently, \[ \alpha_1-2\alpha_2+3\alpha_3=0. \] The coefficients are not all zero, so the four vectors are not linearly independent.