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Prove some relations between given sets of vectors

Let

    \[ S = \{(1,1,1), (0,1,2), (1,0,-1) \}, \quad T = \{ (2,1,0), (2,0,-2) \}, \quad U = \{ (1,2,3), (1,3,5) \}. \]

  1. Prove that L(T) \subseteq L(S).
  2. Determine all of the inclusion relations amongst L(S), L(T), L(U).

  1. Proof. We have

        \begin{align*}  (2,1,0) &= 2 (1,0,-1) + (0,1,2) \\  (2,0,-2) &= 2(1,0,-1). \end{align*}

    Hence, T \subseteq L(S); and so L(T) \subseteq L(S). \qquad \blacksquare

  2. We have from part (a) that L(T) \subseteq L(S). Further,

        \begin{align*}  (1,2,3) &= (1,1,1) + (0,1,2) \\  (1,3,5) &= (1,1,1) + 2(0,1,2). \end{align*}

    Hence, L(U) \subseteq L(S). Finally, L(S) \subseteq L(T) and L(S) \subseteq L(U) since

        \begin{align*}  (1,1,1) &= (2,1,0) - \frac{1}{2}(2,0,-2) \\  (0,1,2) &= (2,1,0) - (2,0,-2) \\  (1,0,-1) &= \frac{1}{2}(2,0,-2) \end{align*}

    and

        \begin{align*}  (1,1,1) &= 2(1,2,3) - (1,3,5) \\  (0,1,2) &= (1,3,5) - (1,2,3) \\  (1,0,-1) &= 3(1,2,3) - 2(1,3,5). \end{align*}

    Therefore, L(S) = L(T) = L(U).

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