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Determine some properties of the vectors A = (1,1,1,0), B = (0,1,1,1), C = (1,1,0,0)

Consider the three vectors in \mathbb{R}^4,

    \[ A = (1,1,1,0), \quad B = (0,1,1,1), \quad C = (1,1,0,0). \]

  1. Determine whether A,B,C are linearly independent or linearly dependent.
  2. Find a nonzero vector D such that A,B,C,D are dependent.
  3. Find a nonzero vector E such that A,B,C,E are independent.
  4. Using the vector E obtained in part (c) express the vector X = (1,2,3,4) as a linear combination of A,B,C,E.

  1. We have

        \begin{align*}  xA + yB + zC = O && \implies && x+z &=0 \\  &&&& x+y+z &= 0 \\  &&&& x+y &= 0 \\  &&&& y &= 0. \end{align*}

    But this implies x = y = z = 0. Hence, A,B,C are linearly independent.

  2. Let D = A = (1,1,1,0). Then, letting c_1 = 1, c_2 = c_3 = 0, and c_4 = -1 we have

        \[ c_1 A + c_2 B + c_3 C + c_4 D = (0,0,0,0). \]

    Hence, A,B,C,D are linearly dependent.

  3. Let E = (0,0,0,1). Then

        \begin{align*}  E = (0,0,0,1) && \implies && c_1 A + c_2 B + c_3 C + c_4 E &= (0,0,0,0) \\  && \implies && c_1 + c_3 &= 0 \\  &&&& c_1 + c_2 + c_4 &= 0 \\  &&&& c_1 + c_2 &= 0 \\  &&&& c_2 + c_4 &= 0. \end{align*}

    These equations implies c_1 = c_2 = c_3 = c_4 = 0. Hence, A,B,C,E are linearly independent.

  4. So we compute

        \begin{align*}  X = c_1 A + c_2 B + c_3 C + c_4 E && \implies && c_1 + c_3 &= 1 \\  && && c_1 + c_2 + c_3 &=2 \\  && && c_1 + c_2 &= 3 \\  && && c_2 + c_4 &= 4. \end{align*}

    Since c_1 + c_2 = 3 (from the third equation), the second implies c_3 = -1. Then the first equation implies c_1 = 2. So the third equation implies c_2 = 1, and finally the fourth equation implies c+4 = 3. Hence,

        \[ X = 2A + B - C + 3E. \]

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