Question

Check if vector u vector v vector w are LI OR LD A)vector u=(1,2,1) vector v=(1,-1,-7) and vector w=(4,5,-4) B)vector u=(1,-1,2) vector v=(-3,4,1) and vector w=(1,0,9) C)vector u=(7,6,1) vector v=(2,0,1) and vector w=(1,-2,1)

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Answer to a math question Check if vector u vector v vector w are LI OR LD A)vector u=(1,2,1) vector v=(1,-1,-7) and vector w=(4,5,-4) B)vector u=(1,-1,2) vector v=(-3,4,1) and vector w=(1,0,9) C)vector u=(7,6,1) vector v=(2,0,1) and vector w=(1,-2,1)

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Jett
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Verifique se vetor \mathbf{u}, vetor \mathbf{v} e vetor \mathbf{w} são LI OU LD

A) vetor \mathbf{u}=(1,2,1), vetor \mathbf{v}=(1,-1,-7) e vetor \mathbf{w}=(4,5,-4)
B) vetor \mathbf{u}=(1,-1,2), vetor \mathbf{v}=(-3,4,1) e vetor \mathbf{w}=(1,0,9)
C) vetor \mathbf{u}=(7,6,1), vetor \mathbf{v}=(2,0,1) e vetor \mathbf{w}=(1,-2,1)

\[Solution\]

A) LI
B) LI
C) LI

\[Step-by-Step\]

A) Para verificar se $\mathbf{u}$, $\mathbf{v}$ e $\mathbf{w}$ são linearmente independentes (LI), montamos a matriz $A$ com os vetores $\mathbf{u}$, $\mathbf{v}$ e $\mathbf{w}$ como colunas e verificamos se a determinante de $A$ é diferente de zero:
A = \begin{pmatrix}1 & 1 & 4 \\2 & -1 & 5 \\1 & -7 & -4\end{pmatrix}
\text{Det}(A) = 1 \cdot (-1 \cdot (-4) - 5 \cdot (-7)) - 1 \cdot (2 \cdot (-4) - 5 \cdot 1) + 4 \cdot (2 \cdot (-7) - (-1) \cdot 1) = 1 \cdot (4 + 35) - 1 \cdot (-8 - 5) + 4 \cdot (-14 + 1) = 1 \cdot 39 + 1 \cdot 13 + 4 \cdot (-13) = 39 + 13 - 52 = 0
\text{Det}(A) = 0 \\
Como o determinante é igual a zero, os vetores são linearmente dependentes.

B) Para verificar se $\mathbf{u}$, $\mathbf{v}$ e $\mathbf{w}$ são linearmente independentes (LI), montamos a matriz $A$ com os vetores $\mathbf{u}$, $\mathbf{v}$ e $\mathbf{w}$ como colunas e verificamos se a determinante de $A$ é diferente de zero:
A = \begin{pmatrix}1 & -3 & 1 \\-1 & 4 & 0 \\2 & 1 & 9\end{pmatrix}
\text{Det}(A) = 1 \cdot (4 \cdot 9 - 1 \cdot 0) - (-3) \cdot (-1 \cdot 9 - 2 \cdot 0) + 1 \cdot (-1 \cdot 1 - 4 \cdot 2) = 1 \cdot (36) - (-3) \cdot (-9) + 1 \cdot (-1 - 8) = 36 - 27 - 9 = 0
\text{Det}(A) = 0 \\
Como o determinante é igual a zero, os vetores são linearmente dependentes.

C) Para verificar se $\mathbf{u}$, $\mathbf{v}$ e $\mathbf{w}$ são linearmente independentes (LI), montamos a matriz $A$ com os vetores $\mathbf{u}$, $\mathbf{v}$ e $\mathbf{w}$ como colunas e verificamos se a determinante de $A$ é diferente de zero:
A = \begin{pmatrix}7 & 2 & 1 \\6 & 0 & -2 \\1 & 1 & 1\end{pmatrix}
\text{Det}(A) = 7 \cdot (0 \cdot 1 - -2 \cdot 1) - 2 \cdot (6 \cdot 1 - 1 \cdot 1) + 1 \cdot (6 \cdot 1 - 0 \cdot 1) = 7 \cdot (2) - 2 \cdot (6 - 1) + 1 \cdot (6) = 14 - 10 + 6 = 10
\text{Det}(A) = 10 \\
Como o determinante é diferente de zero, os vetores são linearmente independentes.

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