Algèbre linéaire Exemples

(u x v) \cdot w = ⎡ ⎢ ⎣ \begin{matrix} a & b & d g & h & j k & l & n \end{matrix} ⎤ ⎥ ⎦

$(u x v) \cdot w = [\begin{array}{c} a & b & d \\ g & h & j \\ k & l & n \end{array}]$

Étape 1

Déterminez la forme d’échelon en ligne réduite.

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Étape 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{a}

$\frac{1}{a}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Étape 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{a}

$\frac{1}{a}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{a}{a} & \frac{b}{a} & \frac{d}{a} g & h & j k & l & n \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{a}{a} & \frac{b}{a} & \frac{d}{a} \\ g & h & j \\ k & l & n \end{array}]$

Étape 1.1.2

Simplifiez

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} g & h & j k & l & n \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ g & h & j \\ k & l & n \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} g & h & j k & l & n \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ g & h & j \\ k & l & n \end{array}]$

Étape 1.2

Perform the row operation

R_{2} = R_{2} - g R_{1}

$R_{2} = R_{2} - g R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Étape 1.2.1

Perform the row operation

R_{2} = R_{2} - g R_{1}

$R_{2} = R_{2} - g R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} g - g \cdot 1 & h - g \frac{b}{a} & j - g \frac{d}{a} k & l & n \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ g - g \cdot 1 & h - g \frac{b}{a} & j - g \frac{d}{a} \\ k & l & n \end{array}]$

Étape 1.2.2

Simplifiez

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & h - \frac{b g}{a} & j - \frac{d g}{a} k & l & n \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ k & l & n \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & h - \frac{b g}{a} & j - \frac{d g}{a} k & l & n \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ k & l & n \end{array}]$

Étape 1.3

Perform the row operation

R_{3} = R_{3} - k R_{1}

$R_{3} = R_{3} - k R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Étape 1.3.1

Perform the row operation

R_{3} = R_{3} - k R_{1}

$R_{3} = R_{3} - k R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & h - \frac{b g}{a} & j - \frac{d g}{a} k - k \cdot 1 & l - k \frac{b}{a} & n - k \frac{d}{a} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ k - k \cdot 1 & l - k \frac{b}{a} & n - k \frac{d}{a} \end{array}]$

Étape 1.3.2

Simplifiez

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & h - \frac{b g}{a} & j - \frac{d g}{a} 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & h - \frac{b g}{a} & j - \frac{d g}{a} 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & h - \frac{b g}{a} & j - \frac{d g}{a} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

Étape 1.4

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{h - \frac{b g}{a}}

$\frac{1}{h - \frac{b g}{a}}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Étape 1.4.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{h - \frac{b g}{a}}

$\frac{1}{h - \frac{b g}{a}}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} \frac{0}{h - \frac{b g}{a}} & \frac{h - \frac{b g}{a}}{h - \frac{b g}{a}} & \frac{j - \frac{d g}{a}}{h - \frac{b g}{a}} 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ \frac{0}{h - \frac{b g}{a}} & \frac{h - \frac{b g}{a}}{h - \frac{b g}{a}} & \frac{j - \frac{d g}{a}}{h - \frac{b g}{a}} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

Étape 1.4.2

Simplifiez

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & l - \frac{b k}{a} & n - \frac{d k}{a} \end{array}]$

Étape 1.5

Perform the row operation

R_{3} = R_{3} - (l - \frac{b k}{a}) R_{2}

$R_{3} = R_{3} - (l - \frac{b k}{a}) R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Étape 1.5.1

Perform the row operation

R_{3} = R_{3} - (l - \frac{b k}{a}) R_{2}

$R_{3} = R_{3} - (l - \frac{b k}{a}) R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 - (l - \frac{b k}{a}) \cdot 0 & l - \frac{b k}{a} - (l - \frac{b k}{a}) \cdot 1 & n - \frac{d k}{a} - (l - \frac{b k}{a}) \frac{a j - d g}{a h - b g} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 - (l - \frac{b k}{a}) \cdot 0 & l - \frac{b k}{a} - (l - \frac{b k}{a}) \cdot 1 & n - \frac{d k}{a} - (l - \frac{b k}{a}) \frac{a j - d g}{a h - b g} \end{array}]$

Étape 1.5.2

Simplifiez

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 & 0 & - \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & 0 & - \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g} \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 & 0 & - \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g} \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & 0 & - \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g} \end{array}]$

Étape 1.6

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j}

$- \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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Étape 1.6.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j}

$- \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} \cdot 0 & - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} \cdot 0 & - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} (- \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g}) \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} \cdot 0 & - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} \cdot 0 & - \frac{a h - b g}{d k h - n a h + n b g + l a j - l d g - b k j} (- \frac{d k h - n a h + n b g + l a j - l d g - b k j}{a h - b g}) \end{array}]$

Étape 1.6.2

Simplifiez

R_{3}

$R_{3}$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & \frac{a j - d g}{a h - b g} 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & \frac{a j - d g}{a h - b g} \\ 0 & 0 & 1 \end{array}]$

Étape 1.7

Perform the row operation

R_{2} = R_{2} - \frac{a j - d g}{a h - b g} R_{3}

$R_{2} = R_{2} - \frac{a j - d g}{a h - b g} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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Étape 1.7.1

Perform the row operation

R_{2} = R_{2} - \frac{a j - d g}{a h - b g} R_{3}

$R_{2} = R_{2} - \frac{a j - d g}{a h - b g} R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 - \frac{a j - d g}{a h - b g} \cdot 0 & 1 - \frac{a j - d g}{a h - b g} \cdot 0 & \frac{a j - d g}{a h - b g} - \frac{a j - d g}{a h - b g} \cdot 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 - \frac{a j - d g}{a h - b g} \cdot 0 & 1 - \frac{a j - d g}{a h - b g} \cdot 0 & \frac{a j - d g}{a h - b g} - \frac{a j - d g}{a h - b g} \cdot 1 \\ 0 & 0 & 1 \end{array}]$

Étape 1.7.2

Simplifiez

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & \frac{d}{a} 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & \frac{d}{a} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Étape 1.8

Perform the row operation

R_{1} = R_{1} - \frac{d}{a} R_{3}

$R_{1} = R_{1} - \frac{d}{a} R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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Étape 1.8.1

Perform the row operation

R_{1} = R_{1} - \frac{d}{a} R_{3}

$R_{1} = R_{1} - \frac{d}{a} R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 - \frac{d}{a} \cdot 0 & \frac{b}{a} - \frac{d}{a} \cdot 0 & \frac{d}{a} - \frac{d}{a} \cdot 1 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - \frac{d}{a} \cdot 0 & \frac{b}{a} - \frac{d}{a} \cdot 0 & \frac{d}{a} - \frac{d}{a} \cdot 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Étape 1.8.2

Simplifiez

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & \frac{b}{a} & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & \frac{b}{a} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Étape 1.9

Perform the row operation

R_{1} = R_{1} - \frac{b}{a} R_{2}

$R_{1} = R_{1} - \frac{b}{a} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Étape 1.9.1

Perform the row operation

R_{1} = R_{1} - \frac{b}{a} R_{2}

$R_{1} = R_{1} - \frac{b}{a} R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 - \frac{b}{a} \cdot 0 & \frac{b}{a} - \frac{b}{a} \cdot 1 & 0 - \frac{b}{a} \cdot 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 - \frac{b}{a} \cdot 0 & \frac{b}{a} - \frac{b}{a} \cdot 1 & 0 - \frac{b}{a} \cdot 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Étape 1.9.2

Simplifiez

R_{1}

$R_{1}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}]$

Étape 2

The pivot positions are the locations with the leading

1

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

a_{11}, a_{22},

$a_{11}, a_{22},$ and

a_{33}

$a_{33}$

Pivot Columns:

1, 2,

$1, 2,$ and

3

$3$

Étape 3

The rank is the number of pivot columns.

3

$3$