((2,0,1,2) , (3,3,-3,3), (1,-1,2,0), (2,2,4,3) ,(0,1,3,0) ) these are vectors , show one vector as a combinaiton of other vectors

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:

\[
\begin{aligned}
2 & = 2a + 3b + c, \\
2 & = 3b - c + d, \\
4 & = a - 3b + 2c + 3d, \\
3 & = 2a + 3b.
\end{aligned}
\]

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.

((2,0,1,2), (3,3,-3,3), (1,-1,2,0), (2,2,4,3), (0,1,3,0)) these are vectors, show one vector as a combination of other vectors

Learn how to express the vector (2,2,4,3) as a linear combination of the vectors (2,0,1,2), (3,3,-3,3), (1,-1,2,0), and (0,1,3,0). This solution involves solving a system of linear equations and understanding vector spaces and linear combinations, suitable for undergraduate linear algebra students.

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