What can any vector x in a vector space be expressed as?

Any vector in a vector space can be expressed as a linear combination of the vectors in its basis because the basis of a vector space provides a minimal set of vectors that span the entire space. This means that any vector within that space can be constructed using a unique combination of those basis vectors, combined with appropriate scalar coefficients.

The concept of a basis is fundamental in linear algebra. For an n-dimensional vector space, a basis consists of n linearly independent vectors, and any vector in the space can be represented as a sum of these basis vectors scaled by their respective coefficients. This property is what defines the structure and dimension of the vector space itself.

While the other options touch on related concepts, they do not encompass the specific and fundamental principle of vector representation in the context of vector spaces. For instance, expressing a vector as a linear combination of non-zero vectors is generally true, but it does not specifically reference the concept of a basis, which is crucial for the precise mathematical framework. Similarly, stating that a vector can be a product of scalars and basis vectors is misleading because vectors are actually sums of scaled basis vectors, not products. Lastly, referring to the coordinates of a vector doesn't capture the full essence of vector representation in terms of linear combinations