If \[ \alpha_1 = (1, -1), \quad \beta_1 = (1, 0), \] \[ \alpha_2 = (2, -1), \quad \beta_2 = (0, 1), \] \[ \alpha_3 = (-3, 2), \quad \beta_3 = (1, 1), \] is there a linear transformation \(T\) from \(\RR^2\) into \(\RR^2\) such that \(T\alpha_i = \beta_i\) for \(i = 1, 2\) and 3?
Solution
No. The vectors satisfy \[ \alpha_3=-\alpha_1-\alpha_2. \] If such a linear transformation existed, linearity would force \[ T\alpha_3=-T\alpha_1-T\alpha_2=-\beta_1-\beta_2=(-1,-1). \] But the prescribed value is \[ \beta_3=(1,1). \] These are unequal in \(\RR^2\), so no such linear transformation exists.