Determine if the given vectors are linearly independent.

**u** = 

−2

0

−3

,    **v** = 

−1

−1

4

,    **w** = 

−4

2

−17

linearly independentlinearly dependent  

To determine if the given vectors **u**, **v**, and **w** are linearly independent or dependent, we need to check if a linear combination of the vectors:

\[
c_1 \mathbf{u} + c_2 \mathbf{v} + c_3 \mathbf{w} = \mathbf{0}
\]

has only the trivial solution \( c_1 = c_2 = c_3 = 0 \).

### The vectors are:
\[
\mathbf{u} = \begin{pmatrix} -2 \\ 0 \\ -3 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} -1 \\ -1 \\ 4 \end{pmatrix}, \quad \mathbf{w} = \begin{pmatrix} -4 \\ 2 \\ -17 \end{pmatrix}
\]

This can be tested by forming a matrix \( A \) with **u**, **v**, and **w** as columns, and determining its determinant. If the determinant of the matrix is non-zero, the vectors are linearly independent. If the determinant is zero, they are linearly dependent.

The matrix \( A \) is:

\[
A = \begin{pmatrix}
-2 & -1 & -4 \\
0 & -1 & 2 \\
-3 & 4 & -17
\end{pmatrix}
\]

Let's calculate the determinant of this matrix.

\[
\text{det}(A) = \begin{vmatrix}
-2 & -1 & -4 \\
0 & -1 & 2 \\
-3 & 4 & -17
\end{vmatrix}
\]

I will compute the determinant now.The determinant of the matrix \( A \) is 0. Since the determinant is zero, the vectors **u**, **v**, and **w** are **linearly dependent**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What does it mean for vectors to be linearly independent in a geometric context?
2. How would you determine the rank of a matrix formed by linearly dependent vectors?
3. Can vectors in a 3D space ever be linearly independent if there are more than three vectors?
4. How does the determinant help in solving systems of linear equations?
5. What are the implications of linearly dependent vectors in real-world applications?

**Tip:** If the determinant of a square matrix is zero, it means the matrix does not have an inverse, and the system of equations it represents does not have a unique solution.

Determine if the given vectors **u** = (-2, 0, -3), **v** = (-1, -1, 4), **w** = (-4, 2, -17) are linearly independent or dependent.

Explore how to determine the linear independence of vectors u = (-2, 0, -3), v = (-1, -1, 4), and w = (-4, 2, -17) using determinants. This problem applies key concepts from linear algebra, including vector spaces, linear combinations, and the determinant theorem. Suitable for high school students and early college learners.

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