Given three vectors v1=(1,3,7),v2=(1,2,0),v3=(1,1,−7)v1=(1,3,7),v2=(1,2,0),v3=(1,1,−7). The vectors v1,v2,v3v1,v2,v3 are linearly independent.

Question 7Answer

True

False

To determine if the vectors \( \mathbf{v_1} = (1, 3, 7) \), \( \mathbf{v_2} = (1, 2, 0) \), and \( \mathbf{v_3} = (1, 1, -7) \) are linearly independent, we need to check whether the only solution to the equation 

\[
c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0}
\]

is \( c_1 = c_2 = c_3 = 0 \). This can be done by forming a matrix with \( \mathbf{v_1} \), \( \mathbf{v_2} \), and \( \mathbf{v_3} \) as columns and checking if the determinant is non-zero.

### Step 1: Form the Matrix

\[
A = \begin{pmatrix}
1 & 1 & 1 \\
3 & 2 & 1 \\
7 & 0 & -7
\end{pmatrix}
\]

### Step 2: Calculate the Determinant

\[
\text{Det}(A) = 1 \cdot \begin{vmatrix} 2 & 1 \\ 0 & -7 \end{vmatrix} - 1 \cdot \begin{vmatrix} 3 & 1 \\ 7 & -7 \end{vmatrix} + 1 \cdot \begin{vmatrix} 3 & 2 \\ 7 & 0 \end{vmatrix}
\]

\[
\text{Det}(A) = 1 \cdot (2 \cdot -7 - 0 \cdot 1) - 1 \cdot (3 \cdot -7 - 1 \cdot 7) + 1 \cdot (3 \cdot 0 - 7 \cdot 2)
\]

\[
\text{Det}(A) = 1 \cdot (-14) - 1 \cdot (-21 - 7) + 1 \cdot (0 - 14)
\]

\[
\text{Det}(A) = -14 + 28 - 14 = 0
\]

### Step 3: Interpret the Result

Since the determinant of the matrix \( A \) is zero, this means the vectors \( \mathbf{v_1} \), \( \mathbf{v_2} \), and \( \mathbf{v_3} \) are **not** linearly independent. Instead, they are linearly dependent.

### Final Answer

- **False**: The vectors \( \mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3} \) are not linearly independent.

Would you like more details on linear dependence, or do you have any other questions?

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Here are five follow-up questions you might consider:

1. What does it mean for vectors to be linearly dependent?
2. How can you determine the rank of a matrix formed by vectors?
3. What is the significance of a zero determinant in the context of vector independence?
4. How can the Gram-Schmidt process be used to orthogonalize vectors?
5. What are some real-world applications where linear independence is crucial?

**Tip:** When vectors are linearly dependent, at least one of the vectors can be expressed as a linear combination of the others.

Learn how to determine if vectors v1 = (1, 3, 7), v2 = (1, 2, 0), and v3 = (1, 1, -7) are linearly independent using determinant calculation. This problem covers key concepts in linear algebra, including vectors, linear independence, and determinants. Suitable for high school students in Grades 10-12.

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