Describe explicitly a linear transformation from \(\RR^3\) into \(\RR^3\) which has as its range the subspace spanned by \((1, 0, -1)\) and \((1, 2, 2)\).
Solution
Let \[ v_1=(1,0,-1), \qquad v_2=(1,2,2). \] Define \[ T(x,y,z)=xv_1+yv_2. \] Explicitly, \[ T(x,y,z)=(x+y,\;2y,\;-x+2y). \] This is linear. Its range is contained in \(\operatorname{span}\{v_1,v_2\}\) by construction. Conversely, \[ T(1,0,0)=v_1, \qquad T(0,1,0)=v_2, \] so both spanning vectors lie in the range. Hence \[ \operatorname{range}T=\operatorname{span}\{(1,0,-1),(1,2,2)\}. \]