Express $- i - 3 j + 4 k$ as the linear combination of the vectors $2 i + j - 4 k, 2 i - j + 3 k$ is $3 i + j - 2 k$

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Solution

$\begin{matrix} - ¯ i - 3 ¯ j + 4 ¯ ¯ ¯ k = x (2 ¯ i + ¯ j - 4 ¯ ¯ ¯ k) + y (2 ¯ i - ¯ j + 3 ¯ ¯ ¯ k) + z (3 ¯ i + ¯ j - 2 ¯ ¯ ¯ k) - ¯ i - 3 ¯ j + 4 ¯ ¯ ¯ k = (2 x + 2 y + 3 z) ¯ i + (x - y + z) ¯ j + (- 4 x + 3 y - z) ¯ ¯ ¯ k 2 x + 2 y + 3 z = - 1 x - y + z = - 3 - 4 x + 3 y - 2 z = 4 \begin{matrix} 2 x + 2 y + 3 z = - 1 2 x - 2 y + 2 z = - 6 \end{matrix} - -------------------- - \begin{matrix} 3 x - 3 y + 3 z = - 9 - 4 x + 3 y - 2 z = 4 \end{matrix} - -------------------- - 4 x + 5 z = - 7 - x + z = - 5 x - z = 5 \begin{matrix} 4 x + 5 z = - 7 4 x - 4 z = 20 \end{matrix} - --------------- - 9 z = - 27 z = - 3 x = 24 + 2 y - 9 = - 1 y = 2 - ¯ i - 3 ¯ j + 4 ¯ ¯ ¯ k = 2 (2 ¯ i + ¯ j - 4 ¯ ¯ ¯ k) + 2 (2 ¯ i - ¯ j + 3 ¯ ¯ ¯ k) - 3 (3 ¯ i + ¯ j - 2 ¯ ¯ ¯ k) \end{matrix}$

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