Final answer:
Vectors that span the kernel of a matrix are found by solving Ax = 0 and can be expressed as a span of vectors corresponding to the matrix's free variables. Column space is spanned by the columns of the matrix, row space by its rows, and orthogonal space consists of all vectors perpendicular to the row space.
Step-by-step explanation:
The student is asking about finding vectors that span specific vector spaces related to a matrix. The kernel, null space, column space, row space, and orthogonal space are different aspects of the structure of a matrix. As for a vector space such as the kernel (or null space) of a matrix, it is the set of all vectors that when multiplied by the matrix, result in the zero vector.
For example, to find vectors that span the kernel of a matrix A, one would solve the homogeneous equation Ax = 0 where x is the vector in question. This requires performing row reduction on matrix A to its reduced row echelon form, identifying the free variables, and then describing the kernel as the span of the vectors corresponding to those free variables.
The column space of a matrix is spanned by its column vectors, and the row space is spanned by its row vectors. Lastly, the orthogonal space of a matrix behaves differently as it comprises all vectors that are perpendicular to the row space of the matrix. This can be found by taking the cross product of the vectors that span the row space, given that they are in three-dimensional space.