Lineare Algebra Beispiele

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

Schritt 1

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

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

Schritt 2

Ermittele die normierte Zeilenstufenform.

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Schritt 2.1

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

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Schritt 2.1.1

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

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

Schritt 2.1.2

Vereinfache $R_{1}$ .

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

Schritt 2.2

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

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Schritt 2.2.1

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

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

Schritt 2.2.2

Vereinfache $R_{2}$ .

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

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 $0$ .

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

Schritt 2.3.2

Vereinfache $R_{3}$ .

$[\begin{array}{c} 1 & - 3 & 0 \\ 0 & 3 & 0 \\ 0 & 11 & 0 \end{array}]$

Schritt 2.4

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

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Schritt 2.4.1

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

$[\begin{array}{c} 1 & - 3 & 0 \\ \frac{0}{3} & \frac{3}{3} & \frac{0}{3} \\ 0 & 11 & 0 \end{array}]$

Schritt 2.4.2

Vereinfache $R_{2}$ .

$[\begin{array}{c} 1 & - 3 & 0 \\ 0 & 1 & 0 \\ 0 & 11 & 0 \end{array}]$

Schritt 2.5

Perform the row operation $R_{3} = R_{3} - 11 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Schritt 2.5.1

Perform the row operation $R_{3} = R_{3} - 11 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & - 3 & 0 \\ 0 & 1 & 0 \\ 0 - 11 \cdot 0 & 11 - 11 \cdot 1 & 0 - 11 \cdot 0 \end{array}]$

Schritt 2.5.2

Vereinfache $R_{3}$ .

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

Schritt 2.6

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

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Schritt 2.6.1

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

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

Schritt 2.6.2

Vereinfache $R_{1}$ .

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

Schritt 3

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

$x = 0$

$y = 0$

$0 = 0$

Schritt 4

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

$[\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 0 \\ 0 \end{array}]$

Schritt 5

Write as a solution set.

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