What is the dimension of a subspace?

The dimension of a subspace relates to the number of linearly independent vectors that span that subspace. In the context of a vector space, the dimension can be understood as the maximum number of linearly independent vectors that can exist within that space, which essentially forms a basis for the subspace.

When considering a subspace within a larger vector space, if there are a certain number of linearly independent constraints that the subspace must satisfy, the dimension of that subspace is computed by taking the dimension of the whole space and subtracting the number of those constraints. This relationship illustrates how the constraints limit the number of independent directions you can take in the subspace compared to the larger space.

This understanding explains why the choice indicating that the dimension of the subspace is calculated by subtracting the number of linearly independent constraints from the dimension of the entire space is accurate. It reflects the foundational principles of linear algebra concerning vector spaces and their subspaces.

Other choices may refer to dimensions or characteristics that do not accurately define the dimensionality of a subspace or do not account for the relationship between the subspace and the larger space effectively.