Give an explicit description of the type (2-25) for the vectors \[ \beta = (b_1, b_2, b_3, b_4, b_5) \] in \(\RR^5\) which are linear combinations of the vectors \[ \alpha_1 = (1, 0, 2, 1, -1), \quad \alpha_2 = (-1, 2, -4, 2, 0), \] \[ \alpha_3 = (2, -1, 5, 2, 1), \quad \alpha_4 = (2, 1, 3, 5, 2). \]
Solution
Let \(W=\operatorname{span}\{\alpha_1,\alpha_2,\alpha_3,\alpha_4\}\). The four given vectors are linearly independent, so \(\dim W=4\). Each one satisfies \[ -2b_1+b_2+b_3=0. \] Hence \[ W\subseteq H:=\{(b_1,\ldots,b_5)\in\RR^5:-2b_1+b_2+b_3=0\}. \] The set \(H\) is the kernel of the nonzero linear functional \[ (b_1,\ldots,b_5)\longmapsto -2b_1+b_2+b_3, \] so \(\dim H=4\). Since \(W\subseteq H\) and both have dimension \(4\), one has \(W=H\).
Therefore the vectors which are linear combinations of the \(\alpha_i\) are exactly those satisfying \[ \boxed{-2b_1+b_2+b_3=0}. \]