determine if b is a linear combination of a1, a2 and a3 a1 = [1;-2;0], a2 = [0;1;2], a3 = [5;-6;8], b= [2;-1;6]

To determine if **b** is a linear combination of **a₁**, **a₂**, and **a₃**, we need to check if there exist scalars \( x_1, x_2, x_3 \) such that:

\[
x_1 \cdot a_1 + x_2 \cdot a_2 + x_3 \cdot a_3 = b
\]

This can be written as the following matrix equation:

\[
x_1 \begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix} + x_2 \begin{bmatrix} 0 \\ 1 \\ 2 \end{bmatrix} + x_3 \begin{bmatrix} 5 \\ -6 \\ 8 \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \\ 6 \end{bmatrix}
\]

This leads to the system of linear equations:

\[
\begin{aligned}
1. & \quad x_1 + 0x_2 + 5x_3 = 2 \\
2. & \quad -2x_1 + x_2 - 6x_3 = -1 \\
3. & \quad 0x_1 + 2x_2 + 8x_3 = 6
\end{aligned}
\]

We will solve this system to determine whether a solution exists for \( x_1, x_2, x_3 \).

Let me solve it.The solution to the system is given by:

\[
x_1 = 2 - 5x_3, \quad x_2 = 3 - 4x_3
\]

This means that there are infinitely many solutions depending on the value of \( x_3 \). Therefore, **b** is a linear combination of \( a_1 \), \( a_2 \), and \( a_3 \), and the scalars \( x_1 \), \( x_2 \), and \( x_3 \) are not unique, but dependent on the value of \( x_3 \).

Would you like more details, or do you have any questions?

Here are five related questions you could explore:
1. How do you express the general form of linear combinations?
2. What is the geometric interpretation of a linear combination of vectors?
3. How can you determine if a set of vectors is linearly dependent?
4. What does it mean when a system of linear equations has infinitely many solutions?
5. Can you find a specific solution by assigning a value to \( x_3 \)?

**Tip:** When a system of equations has infinitely many solutions, it means the vectors involved are not linearly independent!

This problem explores how to determine if vector b = [2; -1; 6] is a linear combination of vectors a1 = [1; -2; 0], a2 = [0; 1; 2], and a3 = [5; -6; 8]. Learn to solve a system of linear equations derived from the matrix equation A * x = b, and understand the concepts of linear dependence and independence. Suitable for undergraduate students studying linear algebra.

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