Final answer:
To determine if B is a linear combination of the columns of matrix A, set up a system of linear equations using the columns as vectors and solve for the scalar multiples. If a non-trivial solution exists, B is indeed a linear combination.
Step-by-step explanation:
To determine if vector B is a linear combination of the columns of matrix A, we must express B as a sum of scalar multiples of the columns of A. We can set up a system of linear equations to solve for the scalars. If the system has a solution, then B can indeed be expressed as a linear combination of the columns. The matrix A is given as:
╔ 1 -6 4 ╗
║ 0 4 3 ║
╚ -3 18 -12 ╝
For B to be a linear combination of the columns of A, there must exist scalars x, y, z such that:
Ax1 + By1 + Cz1 = Bx
Ax2 + By2 + Cz2 = By
Ax3 + By3 + Cz3 = Bz
where x1, y1, z1; x2, y2, z2; x3, y3, z3 are the first, second, and third columns of A respectively, and Bx, By, Bz are the components of vector B.
If the system has a non-trivial solution, then B is a linear combination of the column vectors of matrix A. To find the solution, we could use Gaussian elimination or matrix inversion (if applicable).