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jimmycricket
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Given two subspaces of the vector space of all polynomials of at most degree 3 what is the general method to calculate the intersection of the two subspaces?
The intersection of subspaces refers to the common elements shared between two or more subspaces. It is the set of all vectors that are present in all of the subspaces.
The intersection of subspaces can be calculated by finding the common basis vectors between the subspaces. These basis vectors are then used to create a new subspace, which is the intersection of the original subspaces.
The intersection of subspaces is significant because it can help determine the relationship between different subspaces. If the intersection is non-empty, it means that the subspaces have at least one common vector, and therefore, they are not completely independent.
Yes, the intersection of subspaces can be empty if there are no common elements between the subspaces. This means that the subspaces are completely independent of each other.
The dimension of the intersection of subspaces is always less than or equal to the dimensions of the original subspaces. This is because the intersection can only contain vectors that are present in all of the subspaces, and therefore, it cannot have a higher dimension than any of the original subspaces.