We are given five vectors:

$\mathbf{v_1} = (2, 0, 1, 2), \quad \mathbf{v_2} = (3, 3, -3, 3), \quad \mathbf{v_3} = (1, -1, 2, 0), \quad \mathbf{v_4} = (2, 2, 4, 3), \quad \mathbf{v_5} = (0, 1, 3, 0)$

We need to express one vector as a linear combination of the others. Let’s pick $\mathbf{v_4}$ and attempt to express it as a linear combination of $\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_5}$.

This means we want to find constants $a, b, c, d$ such that:

$\mathbf{v_4} = a \mathbf{v_1} + b \mathbf{v_2} + c \mathbf{v_3} + d \mathbf{v_5}$

In vector form, this equation becomes:

$(2, 2, 4, 3) = a(2, 0, 1, 2) + b(3, 3, -3, 3) + c(1, -1, 2, 0) + d(0, 1, 3, 0)$

This gives a system of four equations:

2 & = 2a + 3b + c, \\ 2 & = 3b - c + d, \\ 4 & = a - 3b + 2c + 3d, \\ 3 & = 2a + 3b. \end{aligned}$$ Now, I will solve this system to find the values of $$a, b, c, d$$. Let me calculate that for you.The solution to the system of equations is: $$a = \frac{5}{3}, \quad b = -\frac{1}{9}, \quad c = -1, \quad d = \frac{4}{3}$$ Thus, we can express $$\mathbf{v_4}$$ as a linear combination of $$\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}, \mathbf{v_5}$$ as: $$\mathbf{v_4} = \frac{5}{3} \mathbf{v_1} - \frac{1}{9} \mathbf{v_2} - \mathbf{v_3} + \frac{4}{3} \mathbf{v_5}$$ This is the desired linear combination. Do you want more details or have any questions? Here are 5 related questions: 1. Can any other vector be written as a combination of the remaining vectors? 2. How can we determine if a set of vectors is linearly independent? 3. What does it mean for a vector to be a linear combination of others? 4. Can this system have more than one solution? 5. What is the geometric interpretation of vectors and their combinations? **Tip:** A set of vectors is linearly dependent if one can be written as a linear combination of the others.