Final answer:
To find a basis for the orthogonal complement to the column space of a matrix, determine the null space of the matrix's transpose, as it consists of all vectors orthogonal to the column space of the original matrix.
Step-by-step explanation:
The subject in question is mathematics, specifically a topic in linear algebra related to vector spaces and orthogonality. To find a basis for the orthogonal complement to the column space of a matrix, we would first determine the column space of the matrix. Once we have the column space, we would then find the vectors that are orthogonal to this space. The set of all such orthogonal vectors would be the orthogonal complement. If the matrix is denoted as A, this can be accomplished by finding the null space of AT (the transpose of the matrix), since the null space of AT consists of all vectors orthogonal to the column space of A.
To find the null space of AT, we perform the following steps:
- Transpose the matrix A to get AT.
- Solve the homogeneous system ATx = 0 to get the set of all solutions x.
- The set of solutions forms the basis for the null space of AT, which is also the orthogonal complement to the column space of A.
The vectors that we find through this process will provide the required basis for the orthogonal complement.