The representation vector of a basis depends on which factor?

The representation vector of a basis indeed depends on the ordering of the elements in the basis. In linear algebra, a basis is defined as a set of vectors that are linearly independent and span a vector space. When constructing a representation vector, you express a vector in terms of the basis vectors. The coefficients in this representation are determined by how the basis vectors are ordered.

For example, consider a basis consisting of vectors ( {b_1, b_2} ). A vector ( v ) can be expressed in terms of this basis as ( v = c_1 b_1 + c_2 b_2 ). If you change the order of the basis to ( {b_2, b_1} ), the representation changes and requires different coefficients, leading to a different representation vector, even though the underlying vector remains the same. Thus, the representation is not merely about the vectors themselves but crucially also about their order in the basis.

In contrast, the number of rows in the matrix pertains more to the dimensions and properties of the matrices rather than the representation of specific vectors. The magnitude of the vector is irrelevant to how a vector is expressed in terms of a basis since representation only concerns the