Final Answer:
The final answer is c1, c2, ..., cn. These vectors form a basis for the column space of matrix C as they are linearly independent and span the column space.Thus,the correct option is d.
Step-by-step explanation:
The basis for the column space of matrix C is formed by the column vectors that are linearly independent and span the column space. Let's denote the column vectors of matrix C as c1, c2, ..., cn. To find a basis for col(C), we need to identify the linearly independent columns of C. This can be done through techniques such as Gaussian elimination or row reduction. Once we have the reduced echelon form of matrix C, the non-zero columns in the original matrix correspond to the basis vectors for col(C).
In the reduced echelon form, the columns with pivots are linearly independent and form the basis for the column space. These basis vectors, c1, c2, ..., cn, are the columns of the original matrix C that correspond to the pivot columns in its reduced echelon form. These vectors are chosen in such a way that any other vector in the column space of C can be expressed as a linear combination of c1, c2, ..., cn. Therefore, they form a basis for the column space of matrix C.
In summary, the basis for col(C) is represented by the vectors c1, c2, ..., cn, where these vectors are the linearly independent columns of matrix C. These basis vectors span the column space of C, providing a set that can express any vector within that space through linear combinations.
Therefore,the correct option is d.