Check whether v1=(1,2,3) v2=(4,1,2) v3=(3,-1,-1)  is linearly independent or dependent

To check if the vectors \( \mathbf{v_1} = (1, 2, 3) \), \( \mathbf{v_2} = (4, 1, 2) \), and \( \mathbf{v_3} = (3, -1, -1) \) are linearly independent or dependent, we can set up the following linear combination equation:

\[
c_1 \mathbf{v_1} + c_2 \mathbf{v_2} + c_3 \mathbf{v_3} = \mathbf{0}
\]

This expands to:

\[
c_1 (1, 2, 3) + c_2 (4, 1, 2) + c_3 (3, -1, -1) = (0, 0, 0)
\]

Which gives the system of equations:

\[
\begin{aligned}
1. \ & c_1 + 4c_2 + 3c_3 = 0 \\
2. \ & 2c_1 + c_2 - c_3 = 0 \\
3. \ & 3c_1 + 2c_2 - c_3 = 0
\end{aligned}
\]

We will solve this system to determine whether there are non-trivial solutions (i.e., solutions where not all the \( c_i \)'s are zero). If such solutions exist, the vectors are linearly dependent; otherwise, they are linearly independent.

I'll solve this now.The solution to the system of equations is:

\[
c_1 = c_3, \quad c_2 = -c_3
\]

This means there is a non-trivial solution where \( c_3 \) can be any scalar, which implies that not all coefficients are zero. Therefore, the vectors \( \mathbf{v_1} \), \( \mathbf{v_2} \), and \( \mathbf{v_3} \) are **linearly dependent**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you interpret the concept of linear dependence in vector spaces?
2. What is the geometric meaning of linearly dependent vectors?
3. Can linearly dependent vectors form a basis for a vector space?
4. How does the rank of a matrix relate to linear independence of vectors?
5. What is the significance of finding non-trivial solutions in such systems?

**Tip:** Linearly dependent vectors lie in the same plane or line in space, meaning one vector can be expressed as a combination of the others.

Check whether v1=(1,2,3) v2=(4,1,2) v3=(3,-1,-1) is linearly independent or dependent

Learn how to determine if the vectors v1=(1,2,3), v2=(4,1,2), and v3=(3,-1,-1) are linearly independent or dependent. This problem involves setting up and solving a system of linear equations and understanding key concepts in linear algebra such as vector spaces and linear combinations. Suitable for students in Grades 11-12 and college.

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