How do I express the idea that a vector space can be represented as a direct sum of two subspaces?
user
to express the idea that a vector space v can be represented as a direct sum of two subspaces u and w, you can state it as:

v = u ⊕ w

this means that the vector space v can be written as a combination of the subspaces u and w, where each vector in v is uniquely represented as the sum of a vector from u and a vector from w. this also implies that the intersection of u and w is the zero vector (the trivial subspace).

in other words, you can think of the direct sum as a way to combine two subspaces in such a manner that their individual structures are preserved, and every vector in the larger vector space v comes from exactly one unique combination of vectors in u and w. this allows you to study the vector space v by analyzing its constituent subspaces u and w.
assistant
To express the idea that a vector space V can be represented as a direct sum of two subspaces U and W, you can state it as:

V = U ⊕ W

This means that the vector space V can be written as a combination of the subspaces U and W, where each vector in V is uniquely represented as the sum of a vector from U and a vector from W. This also implies that the intersection of U and W is the zero vector (the trivial subspace).

In other words, you can think of the direct sum as a way to combine two subspaces in such a manner that their individual structures are preserved, and every vector in the larger vector space V comes from exactly one unique combination of vectors in U and W. This allows you to study the vector space V by analyzing its constituent subspaces U and W.
assistant