Algèbre linéaire Exemples

[\begin{array}{c} 1 & - 1 & 3 \\ 5 & - 4 & - 2 \\ 7 & - 6 & 4 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 \\ 5 & - 4 & - 2 \\ 7 & - 6 & 4 \end{array}]$

Étape 1

Write as an augmented matrix for

A x = 0

$A x = 0$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 5 & - 4 & - 2 & 0 \\ 7 & - 6 & 4 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 5 & - 4 & - 2 & 0 \\ 7 & - 6 & 4 & 0 \end{array}]$

Étape 2

Déterminez la forme d’échelon en ligne réduite.

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Étape 2.1

Perform the row operation

R_{2} = R_{2} - 5 R_{1}

$R_{2} = R_{2} - 5 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Étape 2.1.1

Perform the row operation

R_{2} = R_{2} - 5 R_{1}

$R_{2} = R_{2} - 5 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 5 - 5 \cdot 1 & - 4 - 5 \cdot - 1 & - 2 - 5 \cdot 3 & 0 - 5 \cdot 0 \\ 7 & - 6 & 4 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 5 - 5 \cdot 1 & - 4 - 5 \cdot - 1 & - 2 - 5 \cdot 3 & 0 - 5 \cdot 0 \\ 7 & - 6 & 4 & 0 \end{array}]$

Étape 2.1.2

Simplifiez

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 7 & - 6 & 4 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 7 & - 6 & 4 & 0 \end{array}]$

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 7 & - 6 & 4 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 7 & - 6 & 4 & 0 \end{array}]$

Étape 2.2

Perform the row operation

R_{3} = R_{3} - 7 R_{1}

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Étape 2.2.1

Perform the row operation

R_{3} = R_{3} - 7 R_{1}

$R_{3} = R_{3} - 7 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 7 - 7 \cdot 1 & - 6 - 7 \cdot - 1 & 4 - 7 \cdot 3 & 0 - 7 \cdot 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 7 - 7 \cdot 1 & - 6 - 7 \cdot - 1 & 4 - 7 \cdot 3 & 0 - 7 \cdot 0 \end{array}]$

Étape 2.2.2

Simplifiez

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 1 & - 17 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 1 & - 17 & 0 \end{array}]$

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 1 & - 17 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 1 & - 17 & 0 \end{array}]$

Étape 2.3

Perform the row operation

R_{3} = R_{3} - R_{2}

$R_{3} = R_{3} - R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Étape 2.3.1

Perform the row operation

R_{3} = R_{3} - R_{2}

$R_{3} = R_{3} - R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 - 0 & 1 - 1 & - 17 + 17 & 0 - 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 - 0 & 1 - 1 & - 17 + 17 & 0 - 0 \end{array}]$

Étape 2.3.2

Simplifiez

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

$[\begin{array}{c} 1 & - 1 & 3 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Étape 2.4

Perform the row operation

R_{1} = R_{1} + R_{2}

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

1, 2

$1, 2$ a

0

$0$ .

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Étape 2.4.1

Perform the row operation

R_{1} = R_{1} + R_{2}

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

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 3 - 17 & 0 + 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 3 - 17 & 0 + 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]$

Étape 2.4.2

Simplifiez

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 & - 14 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

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

[\begin{array}{c} 1 & 0 & - 14 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

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

[\begin{array}{c} 1 & 0 & - 14 & 0 \\ 0 & 1 & - 17 & 0 \\ 0 & 0 & 0 & 0 \end{array}]

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

Étape 3

Use the result matrix to declare the final solution to the system of equations.

x - 14 z = 0

$x - 14 z = 0$

y - 17 z = 0

$y - 17 z = 0$

0 = 0

$0 = 0$

Étape 4

Write a solution vector by solving in terms of the free variables in each row.

[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 14 z \\ 17 z \\ z \end{array}]

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 14 z \\ 17 z \\ z \end{array}]$

Étape 5

Write the solution as a linear combination of vectors.

[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 14 \\ 17 \\ 1 \end{array}]

$[\begin{array}{c} x \\ y \\ z \end{array}] = z [\begin{array}{c} 14 \\ 17 \\ 1 \end{array}]$

Étape 6

Write as a solution set.

{z [\begin{array}{c} 14 \\ 17 \\ 1 \end{array}] | z \in R}

${z [\begin{array}{c} 14 \\ 17 \\ 1 \end{array}] | z \in R}$