1. **State the problem:** Compute the linear combination $u - 2v + 4w$ where $u = [1, 2, 1, 0]$, $v = [-2, 0, 1, 6]$, $w = [3, -5, |, -2]$. Note: The vector $w$ contains a vertical bar $|$ which is not a valid number. Assuming it is a typo and should be a number, let's replace $|$ with $0$ for calculation purposes: $w = [3, -5, 0, -2]$. 2. **Recall the formula for linear combination:** $$a u + b v + c w = [a u_1 + b v_1 + c w_1, a u_2 + b v_2 + c w_2, a u_3 + b v_3 + c w_3, a u_4 + b v_4 + c w_4]$$ 3. **Apply the coefficients:** $$u - 2v + 4w = 1 \cdot u + (-2) \cdot v + 4 \cdot w$$ 4. **Calculate each component:** - First component: $1 \times 1 + (-2) \times (-2) + 4 \times 3 = 1 + 4 + 12 = 17$ - Second component: $1 \times 2 + (-2) \times 0 + 4 \times (-5) = 2 + 0 - 20 = -18$ - Third component: $1 \times 1 + (-2) \times 1 + 4 \times 0 = 1 - 2 + 0 = -1$ - Fourth component: $1 \times 0 + (-2) \times 6 + 4 \times (-2) = 0 - 12 - 8 = -20$ 5. **Final answer:** $$[17, -18, -1, -20]$$ This is the resulting vector from the linear combination $u - 2v + 4w$ assuming the third component of $w$ is $0$.