That doesn't make any sense. Instead of guessing, please tell us what F[a, b] means relative to this problem. It should say in the problem itself.slugbunny said:
In linear algebra, subspaces refer to a subset of a vector space that is closed under vector addition and scalar multiplication. This means that if you take any two vectors from the subspace and add them together, the resulting vector will still be in the subspace. Similarly, multiplying a vector in the subspace by a scalar will also result in a vector that is still within the subspace.
In order for a set of vectors to be considered a subspace, it must meet three conditions: it must contain the zero vector, it must be closed under vector addition, and it must be closed under scalar multiplication. This means that if you take any two vectors in the set and add them together or multiply them by a scalar, the resulting vector must also be in the set.
One example of a subspace is the set of all 2-dimensional vectors with real number components. This set contains the zero vector, is closed under vector addition (e.g. (1,2) + (3,4) = (4,6) which is still a 2-dimensional vector), and is closed under scalar multiplication (e.g. 2(1,2) = (2,4) which is still a 2-dimensional vector).
A span is the set of all possible linear combinations of a given set of vectors. It can include vectors that are not necessarily in the original set. A subspace, on the other hand, must contain the original set of vectors and must also be closed under addition and scalar multiplication.
Subspaces are used in a variety of real-world applications, including data analysis, computer graphics, and machine learning. In data analysis, subspaces are used to represent patterns and relationships within a dataset. In computer graphics, subspaces are used to create 3-dimensional images and animations. In machine learning, subspaces are used to reduce the dimensionality of data, making it easier to analyze and classify.