Algebra lineare Esempi

$s = {[\begin{array}{c} 2 \\ - 1 \\ 3 \\ 4 \\ 2 \end{array}], [\begin{array}{c} 6 \\ 2 \\ - 1 \\ 3 \\ 4 \end{array}]}$

Passaggio 1

To determine if the columns in the matrix are linearly dependent, determine if the equation $A x = 0$ has any non-trivial solutions.

Passaggio 2

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

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

Passaggio 3

Trova la forma ridotta a scala per righe di Echelon.

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Passaggio 3.1

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

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Passaggio 3.1.1

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

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

Passaggio 3.1.2

Semplifica $R_{1}$ .

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

Passaggio 3.2

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

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Passaggio 3.2.1

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

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

Passaggio 3.2.2

Semplifica $R_{2}$ .

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

Passaggio 3.3

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

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Passaggio 3.3.1

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

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

Passaggio 3.3.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 4 & 3 & 0 \\ 2 & 4 & 0 \end{array}]$

Passaggio 3.4

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

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Passaggio 3.4.1

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

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 4 - 4 \cdot 1 & 3 - 4 \cdot 3 & 0 - 4 \cdot 0 \\ 2 & 4 & 0 \end{array}]$

Passaggio 3.4.2

Semplifica $R_{4}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 2 & 4 & 0 \end{array}]$

Passaggio 3.5

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

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Passaggio 3.5.1

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

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 2 - 2 \cdot 1 & 4 - 2 \cdot 3 & 0 - 2 \cdot 0 \end{array}]$

Passaggio 3.5.2

Semplifica $R_{5}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 5 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Passaggio 3.6

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

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Passaggio 3.6.1

Multiply each element of $R_{2}$ by $\frac{1}{5}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 3 & 0 \\ \frac{0}{5} & \frac{5}{5} & \frac{0}{5} \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Passaggio 3.6.2

Semplifica $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & - 10 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Passaggio 3.7

Perform the row operation $R_{3} = R_{3} + 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Passaggio 3.7.1

Perform the row operation $R_{3} = R_{3} + 10 R_{2}$ to make the entry at $3, 2$ a $0$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 + 10 \cdot 0 & - 10 + 10 \cdot 1 & 0 + 10 \cdot 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Passaggio 3.7.2

Semplifica $R_{3}$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & - 9 & 0 \\ 0 & - 2 & 0 \end{array}]$

Passaggio 3.8

Perform the row operation $R_{4} = R_{4} + 9 R_{2}$ to make the entry at $4, 2$ a $0$ .

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Passaggio 3.8.1

Perform the row operation $R_{4} = R_{4} + 9 R_{2}$ to make the entry at $4, 2$ a $0$ .

$[\begin{array}{c} 1 & 3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 + 9 \cdot 0 & - 9 + 9 \cdot 1 & 0 + 9 \cdot 0 \\ 0 & - 2 & 0 \end{array}]$

Passaggio 3.8.2

Semplifica $R_{4}$ .

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

Passaggio 3.9

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

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Passaggio 3.9.1

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

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

Passaggio 3.9.2

Semplifica $R_{5}$ .

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

Passaggio 3.10

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

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Passaggio 3.10.1

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

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

Passaggio 3.10.2

Semplifica $R_{1}$ .

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

Passaggio 4

Remove rows that are all zeros.

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

Passaggio 5

Write the matrix as a system of linear equations.

$x = 0$

$y = 0$

Passaggio 6

Since the only solution to $A x = 0$ is the trivial solution, the vectors are linearly independent.

Linearmente indipendente