Final answer:
To determine if a vector is in the set spanned by the columns of a matrix, set up a linear combination equation with scalars and solve for the values.
Step-by-step explanation:
To determine if a vector, v, is in the set spanned by the columns of matrix, b, we can use the concept of linear combinations. If vector v can be written as a linear combination of the columns of matrix b, then it is in the set spanned by the columns of b.
To check this, set up the equation v = cx1 + dx2 + ex3 + fx4, where c, d, e, and f are scalars and x1, x2, x3, and x4 are the columns of matrix b. Solve this equation to find the values of c, d, e, and f. If there exists a solution, then v is in the set spanned by the columns of b.
For the given vector v = [-7, 1, 24, 16], we need to solve the equation [-7, 1, 24, 16] = c[1, 2, 3, 4] + d[5, 6, 7, 8] + e[9, 10, 11, 12] + f[13, 14, 15, 16]. By solving this system of equations, we can determine if such values of c, d, e, and f exist.