Lineare Algebra Beispiele

$[\begin{array}{c} \sqrt{3} & 7 & 2 & 1 \\ 5 & 9 & 15 & 8 \\ 4 & 6 & 7 & 6 \end{array}]$

Schritt 1

Write as an augmented matrix for $A x = 0$ .

$[\begin{array}{c} \sqrt{3} & 7 & 2 & 1 & 0 \\ 5 & 9 & 15 & 8 & 0 \\ 4 & 6 & 7 & 6 & 0 \end{array}]$

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

Multiply each element of $R_{1}$ by $\frac{1}{\sqrt{3}}$ to make the entry at $1, 1$ a $1$ .

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Schritt 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{\sqrt{3}}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{\sqrt{3}}{\sqrt{3}} & \frac{7}{\sqrt{3}} & \frac{2}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{0}{\sqrt{3}} \\ 5 & 9 & 15 & 8 & 0 \\ 4 & 6 & 7 & 6 & 0 \end{array}]$

Schritt 2.1.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 5 & 9 & 15 & 8 & 0 \\ 4 & 6 & 7 & 6 & 0 \end{array}]$

Schritt 2.2

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Schritt 2.2.1

Perform the row operation $R_{2} = R_{2} - 5 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 5 - 5 \cdot 1 & 9 - 5 \frac{7 \sqrt{3}}{3} & 15 - 5 \frac{2 \sqrt{3}}{3} & 8 - 5 \frac{\sqrt{3}}{3} & 0 - 5 \cdot 0 \\ 4 & 6 & 7 & 6 & 0 \end{array}]$

Schritt 2.2.2

Vereinfache $R_{2}$ .

Schritt 2.3

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Schritt 2.3.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 9 - \frac{35 \sqrt{3}}{3} & 15 - \frac{10 \sqrt{3}}{3} & 8 - \frac{5 \sqrt{3}}{3} & 0 \\ 4 - 4 \cdot 1 & 6 - 4 \frac{7 \sqrt{3}}{3} & 7 - 4 \frac{2 \sqrt{3}}{3} & 6 - 4 \frac{\sqrt{3}}{3} & 0 - 4 \cdot 0 \end{array}]$

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 9 - \frac{35 \sqrt{3}}{3} & 15 - \frac{10 \sqrt{3}}{3} & 8 - \frac{5 \sqrt{3}}{3} & 0 \\ 0 & 6 - \frac{28 \sqrt{3}}{3} & 7 - \frac{8 \sqrt{3}}{3} & 6 - \frac{4 \sqrt{3}}{3} & 0 \end{array}]$

Schritt 2.4

Multiply each element of $R_{2}$ by $\frac{1}{9 - \frac{35 \sqrt{3}}{3}}$ to make the entry at $2, 2$ a $1$ .

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Schritt 2.4.1

Multiply each element of $R_{2}$ by $\frac{1}{9 - \frac{35 \sqrt{3}}{3}}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ \frac{0}{9 - \frac{35 \sqrt{3}}{3}} & \frac{9 - \frac{35 \sqrt{3}}{3}}{9 - \frac{35 \sqrt{3}}{3}} & \frac{15 - \frac{10 \sqrt{3}}{3}}{9 - \frac{35 \sqrt{3}}{3}} & \frac{8 - \frac{5 \sqrt{3}}{3}}{9 - \frac{35 \sqrt{3}}{3}} & \frac{0}{9 - \frac{35 \sqrt{3}}{3}} \\ 0 & 6 - \frac{28 \sqrt{3}}{3} & 7 - \frac{8 \sqrt{3}}{3} & 6 - \frac{4 \sqrt{3}}{3} & 0 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 1 & - \frac{55 + 435 \sqrt{3}}{982} & - \frac{41 + 235 \sqrt{3}}{982} & 0 \\ 0 & 6 - \frac{28 \sqrt{3}}{3} & 7 - \frac{8 \sqrt{3}}{3} & 6 - \frac{4 \sqrt{3}}{3} & 0 \end{array}]$

Schritt 2.5

Perform the row operation $R_{3} = R_{3} - (6 - \frac{28 \sqrt{3}}{3}) R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 2.5.1

Perform the row operation $R_{3} = R_{3} - (6 - \frac{28 \sqrt{3}}{3}) R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 1 & - \frac{55 + 435 \sqrt{3}}{982} & - \frac{41 + 235 \sqrt{3}}{982} & 0 \\ 0 - (6 - \frac{28 \sqrt{3}}{3}) \cdot 0 & 6 - \frac{28 \sqrt{3}}{3} - (6 - \frac{28 \sqrt{3}}{3}) \cdot 1 & 7 - \frac{8 \sqrt{3}}{3} - (6 - \frac{28 \sqrt{3}}{3}) (- \frac{55 + 435 \sqrt{3}}{982}) & 6 - \frac{4 \sqrt{3}}{3} - (6 - \frac{28 \sqrt{3}}{3}) (- \frac{41 + 235 \sqrt{3}}{982}) & 0 - (6 - \frac{28 \sqrt{3}}{3}) \cdot 0 \end{array}]$

Schritt 2.5.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 1 & - \frac{55 + 435 \sqrt{3}}{982} & - \frac{41 + 235 \sqrt{3}}{982} & 0 \\ 0 & 0 & - \frac{2488 + 261 \sqrt{3}}{491} & - \frac{221 + 141 \sqrt{3}}{491} & 0 \end{array}]$

Schritt 2.6

Multiply each element of $R_{3}$ by $- \frac{491}{2488 + 261 \sqrt{3}}$ to make the entry at $3, 3$ a $1$ .

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Schritt 2.6.1

Multiply each element of $R_{3}$ by $- \frac{491}{2488 + 261 \sqrt{3}}$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 1 & - \frac{55 + 435 \sqrt{3}}{982} & - \frac{41 + 235 \sqrt{3}}{982} & 0 \\ - \frac{491}{2488 + 261 \sqrt{3}} \cdot 0 & - \frac{491}{2488 + 261 \sqrt{3}} \cdot 0 & - \frac{491}{2488 + 261 \sqrt{3}} (- \frac{2488 + 261 \sqrt{3}}{491}) & - \frac{491}{2488 + 261 \sqrt{3}} (- \frac{221 + 141 \sqrt{3}}{491}) & - \frac{491}{2488 + 261 \sqrt{3}} \cdot 0 \end{array}]$

Schritt 2.6.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 1 & - \frac{55 + 435 \sqrt{3}}{982} & - \frac{41 + 235 \sqrt{3}}{982} & 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 2.7

Perform the row operation $R_{2} = R_{2} + \frac{55 + 435 \sqrt{3}}{982} R_{3}$ to make the entry at $2, 3$ a $0$ .

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Schritt 2.7.1

Perform the row operation $R_{2} = R_{2} + \frac{55 + 435 \sqrt{3}}{982} R_{3}$ to make the entry at $2, 3$ a $0$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 + \frac{55 + 435 \sqrt{3}}{982} \cdot 0 & 1 + \frac{55 + 435 \sqrt{3}}{982} \cdot 0 & - \frac{55 + 435 \sqrt{3}}{982} + \frac{55 + 435 \sqrt{3}}{982} \cdot 1 & - \frac{41 + 235 \sqrt{3}}{982} + \frac{55 + 435 \sqrt{3}}{982} \cdot \frac{895 + 597 \sqrt{3}}{12191} & 0 + \frac{55 + 435 \sqrt{3}}{982} \cdot 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 2.7.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & \frac{2 \sqrt{3}}{3} & \frac{\sqrt{3}}{3} & 0 \\ 0 & 1 & 0 & \frac{669 - 4975 \sqrt{3}}{24382} & 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 2.8

Perform the row operation $R_{1} = R_{1} - \frac{2 \sqrt{3}}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

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Schritt 2.8.1

Perform the row operation $R_{1} = R_{1} - \frac{2 \sqrt{3}}{3} R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 - \frac{2 \sqrt{3}}{3} \cdot 0 & \frac{7 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \cdot 0 & \frac{2 \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \cdot 1 & \frac{\sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \cdot \frac{895 + 597 \sqrt{3}}{12191} & 0 - \frac{2 \sqrt{3}}{3} \cdot 0 \\ 0 & 1 & 0 & \frac{669 - 4975 \sqrt{3}}{24382} & 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 2.8.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & \frac{7 \sqrt{3}}{3} & 0 & \frac{3467 \sqrt{3} - 1194}{12191} & 0 \\ 0 & 1 & 0 & \frac{669 - 4975 \sqrt{3}}{24382} & 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 2.9

Perform the row operation $R_{1} = R_{1} - \frac{7 \sqrt{3}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

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Schritt 2.9.1

Perform the row operation $R_{1} = R_{1} - \frac{7 \sqrt{3}}{3} R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - \frac{7 \sqrt{3}}{3} \cdot 0 & \frac{7 \sqrt{3}}{3} - \frac{7 \sqrt{3}}{3} \cdot 1 & 0 - \frac{7 \sqrt{3}}{3} \cdot 0 & \frac{3467 \sqrt{3} - 1194}{12191} - \frac{7 \sqrt{3}}{3} \cdot \frac{669 - 4975 \sqrt{3}}{24382} & 0 - \frac{7 \sqrt{3}}{3} \cdot 0 \\ 0 & 1 & 0 & \frac{669 - 4975 \sqrt{3}}{24382} & 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 2.9.2

Vereinfache $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & \frac{32437 + 5373 \sqrt{3}}{24382} & 0 \\ 0 & 1 & 0 & \frac{669 - 4975 \sqrt{3}}{24382} & 0 \\ 0 & 0 & 1 & \frac{895 + 597 \sqrt{3}}{12191} & 0 \end{array}]$

Schritt 3

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

$x_{1} + \frac{32437 + 5373 \sqrt{3}}{24382} x_{4} = 0$

$x_{2} + \frac{669 - 4975 \sqrt{3}}{24382} x_{4} = 0$

$x_{3} + \frac{895 + 597 \sqrt{3}}{12191} x_{4} = 0$

Schritt 4

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

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = [\begin{array}{c} - \frac{32437 x_{4}}{24382} - \frac{5373 \sqrt{3} x_{4}}{24382} \\ - \frac{669 x_{4}}{24382} + \frac{4975 \sqrt{3} x_{4}}{24382} \\ - \frac{895 x_{4}}{12191} - \frac{597 \sqrt{3} x_{4}}{12191} \\ x_{4} \end{array}]$

Schritt 5

Write the solution as a linear combination of vectors.

$[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}] = x_{4} [\begin{array}{c} - \frac{32437}{24382} - \frac{5373 \sqrt{3}}{24382} \\ - \frac{669}{24382} + \frac{4975 \sqrt{3}}{24382} \\ - \frac{895}{12191} - \frac{597 \sqrt{3}}{12191} \\ 1 \end{array}]$

Schritt 6

Write as a solution set.

${x_{4} [\begin{array}{c} - \frac{32437}{24382} - \frac{5373 \sqrt{3}}{24382} \\ - \frac{669}{24382} + \frac{4975 \sqrt{3}}{24382} \\ - \frac{895}{12191} - \frac{597 \sqrt{3}}{12191} \\ 1 \end{array}] | x_{4} \in R}$