What defines a spanning set of matrices?

A spanning set of matrices is defined by the property that any matrix in the corresponding vector space can be expressed as a linear combination of matrices from that set. In other words, if you have a spanning set of matrices, it means that you can create any matrix in that space by combining the matrices in the set with appropriate coefficients.

So, when considering why the correct answer is about any matrix being representable as a linear combination of the matrices in the set, this aligns perfectly with the definition of a spanning set. It highlights the ability of the set to cover the entire space of matrices, which is the essence of what spanning entails in linear algebra.

In contrast, while diagonalizable forms, the dimensions of matrices, and the inclusion of zero matrices are important aspects of linear algebra, they do not represent the fundamental definition of a spanning set. A spanning set does not require the specifics of diagonalization or restrictions on dimensions; it simply needs to have the capacity to generate any matrix through linear combinations. This underlying concept is key for understanding the role of spanning sets in vector spaces.