Final Answer:
To determine if a vector v is in the set spanned by the columns of matrix B, we can either solve the equation Bc = v using row echelon form or check if v is in the column space of B by reducing the augmented matrix [B | v] to row echelon form.
Explanation:
To determine if a vector v is in the set spanned by the columns of matrix B, we need to check if there exist scalars c1, c2, ..., cn such that v can be expressed as a linear combination of the columns of B. In other words, we need to check if the equation Bc = v has a solution for some vector c.
One way to check if the equation Bc = v has a solution is to form the augmented matrix [B | v] and reduce it to row echelon form using elementary row operations. If the resulting matrix has a row of the form [0 0 ... 0 | d], where d is nonzero, then the equation has no solution, and therefore v is not in the set spanned by the columns of B. Otherwise, the equation has at least one solution, and v is in the set spanned by the columns of B.
Alternatively, we can use the fact that the columns of B span the column space of B, which is the set of all linear combinations of the columns of B. Therefore, v is in the set spanned by the columns of B if and only if v is in the column space of B. To check if v is in the column space of B, we can form the augmented matrix [B | v] and reduce it to row echelon form using elementary row operations. If the resulting matrix has a pivot in every row, then v is in the column space of B, and therefore v is in the set spanned by the columns of B. Otherwise, v is not in the column space of B, and therefore v is not in the set spanned by the columns of B.
In summary, to determine if a vector v is in the set spanned by the columns of matrix B, we can either solve the equation Bc = v using row echelon form or check if v is in the column space of B by reducing the augmented matrix [B | v] to row echelon form.