To determine the value of $k$ such that the vector $\mathbf{v} = (1, -2, k)$ in $\mathbb{R}^3$ is a linear combination of the vectors $\mathbf{u} = (3, 0, -2)$ and $\mathbf{w} = (2, -1, -5)$, we solve the equation:

$\mathbf{v} = a\mathbf{u} + b\mathbf{w}$

This expands to:

$(1, -2, k) = a(3, 0, -2) + b(2, -1, -5)$

Equating components gives the system of equations:

1 = 3a + 2b \tag{1} -2 = -b \tag{2} k = -2a - 5b \tag{3}

Step 1: Solve for $b$ from Equation (2)

From $-2 = -b$, we get:

$b = 2$

Step 2: Substitute $b = 2$ into Equation (1)

$1 = 3a + 2(2)$ $1 = 3a + 4$ $3a = -3 \implies a = -1$

Step 3: Substitute $a = -1$ and $b = 2$ into Equation (3)

$k = -2(-1) - 5(2)$ $k = 2 - 10$ $k = -8$

Final Answer:

The value of $k$ is $-8$.

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Would you like a detailed breakdown of each step or additional clarifications?

Related Questions:

1. How can you verify if a vector is a linear combination of given vectors?
2. What happens if the given vectors are linearly dependent?
3. How does this method extend to higher dimensions?
4. Can this problem be solved using matrix row reduction? How?
5. How does the value of $k$ affect the dependency of the vectors?

Tip:

Always check if the given vectors are linearly independent before solving, as dependency affects the uniqueness of the solution.