الجبر الخطي الأمثلة

$[\begin{array}{c} 1 & 1 & 3 & 0 & 2 \\ 4 & 5 & - 2 & 6 & 0 \end{array}]$

خطوة 1

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 1.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

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خطوة 1.1.1

Perform the row operation $R_{2} = R_{2} - 4 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 3 & 0 & 2 \\ 4 - 4 \cdot 1 & 5 - 4 \cdot 1 & - 2 - 4 \cdot 3 & 6 - 4 \cdot 0 & 0 - 4 \cdot 2 \end{array}]$

خطوة 1.1.2

بسّط $R_{2}$ .

$[\begin{array}{c} 1 & 1 & 3 & 0 & 2 \\ 0 & 1 & - 14 & 6 & - 8 \end{array}]$

خطوة 1.2

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

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خطوة 1.2.1

Perform the row operation $R_{1} = R_{1} - R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 0 & 1 - 1 & 3 + 14 & 0 - 6 & 2 + 8 \\ 0 & 1 & - 14 & 6 & - 8 \end{array}]$

خطوة 1.2.2

بسّط $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 17 & - 6 & 10 \\ 0 & 1 & - 14 & 6 & - 8 \end{array}]$

خطوة 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$