What is the dimension of a vector space represented as?

The dimension of a vector space is defined as the maximum number of linearly independent vectors that can exist within that space, which is directly related to the concept of a basis. A basis is a set of vectors that both spans the vector space and is linearly independent. Therefore, the dimension can also be understood through another perspective, where it is the number of vectors in any basis of the vector space.

When we refer to the maximum number of vectors in any basis, we are confirming that this number represents the "size" of the space in terms of expressibility: how many directions or axes of movement exist within that space. Each basis provides a unique way to represent every vector in that space, so knowing the dimension allows for understanding how complex or multidimensional the space is.

The other options do not accurately reflect the definition of dimension: the number of linearly independent vectors is related to dimension but is not a comprehensive definition; the sum of all vector coordinates does not relate to the dimension in any way; and the product of dimensions of subspaces does not apply directly to defining the dimension of the vector space itself. Therefore, the concept of dimension being equated to the maximum number of vectors in any basis provides a clear and essential understanding of