Question

Determine whether [6,10,2] ᵀ is a linear combination of [1,3,2]ᵀ, [2,8,−1]ᵀ, and [−1,9,2]ᵀ.

Answer 1

To determine if [6,10,2]ᵀ is a linear combination of [1,3,2]ᵀ, [2,8,−1]ᵀ, and [−1,9,2]ᵀ, we set up and solve a system of linear equations to find the coefficients that combine the given vectors to form the target vector.

Step-by-step explanation:

The question asks us to determine whether the vector [6,10,2]ᵀ is a linear combination of the vectors [1,3,2]ᵀ, [2,8,−1]ᵀ, and [−1,9,2]ᵀ. To find out if one vector is a linear combination of others, we need to solve a system of linear equations that represents the coefficients that multiply each of the given vectors to obtain the target vector.

1. Set up the matrix equation of the form Ax=b, where A is the matrix whose columns are the vectors we are given, x is the column vector of coefficients we are solving for, and b is our target vector [6,10,2]ᵀ.
2. Solve the equation using elimination, substitution, or matrix methods such as Gauss-Jordan elimination or applying the inverse of the matrix A if it exists.
3. Check if the obtained solution is valid, which would determine if [6,10,2]ᵀ is a linear combination of the other vectors.