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Prove some properties of an alternate definition of the dot product

Instead of the usual definition of the dot product, suppose we define the dot product of A = (a_1, \ldots, a_n) and B = (b_1, \ldots, b_n) by the formula

    \[ A \cdot B = \sum_{k=1}^n | a_k b_k |. \]

Which of the properties of Theorem 12.2 (on page 451 of Apostol) are still valid under this new definition? Is the Cauchy-Schwarz inequality still valid?


  1. Property (a) is still valid since

        \[ A \cdot B = \sum_{k=1}^n | a_k b_k | = \sum_{k=1}^n | b_k a_k| = B \cdot A. \]

  2. Property (b) fails since

        \[ A \cdot (B+C) = \sum_{k=1}^n | a_k (b_k +c_k) | = \sum_{k=1}^n | a_k b_k + a_k c_k | \]

    while

        \[ A \cdot B + A \cdot C = \sum_{k=1}^n |a_k b_k| + \sum_{k=1}^n |a_k c_k| = \sum_{k=1}^n ( a_k b_k + a_k c_k). \]

    But, | a_k b_k + a_k c_k| is not necessarily equal to |a_k b_k| + |a_k c_k|.

  3. Property (c) is still valid since

        \begin{align*}  c(A \cdot B) &= c \sum_{k=1}^n |a_k b_k| \\  &= \sum_{k=1}^n c|a_k| |b_k|  = (cA) \cdot B \\  &= \sum_{k=1}^n |a_k c|b_k| = A \cdot (cB). \end{align*}

  4. Property (d) is still valid since

        \[ A \cdot A = \sum_{k=1}^n |a_k^2| = \sum_{k=1}^n a_k^2 > 0 \]

    if A \neq O.

  5. Property (e) is still valid since

        \[ A \cdot A = \sum_{k=1}^n |a_k^2| = \sum_{k=1}^n 0 = 0 \]

    if A = O.

  6. The Cauchy-Schwartz inequality still holds since the formula is identical after squaring.

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