Question

Determine if b = 2 -1 6 is a linear combination of a1 = 1 -2 0, a2 = 5 -6 8, and a3 = 0 1 2?

Answer 1

To determine if b is a linear combination of a1, a2, and a3, set up and solve a system of equations using x, y, and z as scalars for the vectors. If a solution exists, b is a linear combination of the given vectors; if not, it is not.

Step-by-step explanation:

To determine if b = 2 -1 6 is a linear combination of a1 = 1 -2 0, a2 = 5 -6 8, and a3 = 0 1 2, we need to see if there exist scalars x, y, and z such that x*a1 + y*a2 + z*a3 = b. This can be set up as a matrix equation to solve for x, y, and z.

Setting up the equation gives us:

• x(1) + y(5) + z(0) = 2
• x(-2) + y(-6) + z(1) = -1
• x(0) + y(8) + z(2) = 6

This translates to the following system of equations:

1. x + 5y = 2
2. -2x - 6y + z = -1
3. 8y + 2z = 6

We solve this system of equations to find the values of x, y, and z. If a solution exists, then b is a linear combination of a1, a2, and a3. Otherwise, b cannot be written as a linear combination of these vectors.