Yes, because ##\dim(W1+W2) + \dim(W1 \cap W2) = \dim(W1) + \dim(W2)##.La_Lune said:Hi all,
Say that I already know W1, W2 are both subspaces of a vector space V, W1∩W2={0}, and that dim(W1)+dim(W2)=dim(V)=n, can I thus conclude that V=W1+W2, namely V is the direct sum of W1 and W2?
When we say that V is a direct sum of W1 and W2, it means that V is the direct sum of two subspaces, W1 and W2, where every vector in V can be uniquely expressed as the sum of a vector in W1 and a vector in W2.
In order to prove that V is a direct sum of W1 and W2, we need to show that the intersection of W1 and W2 is only the zero vector and that the sum of W1 and W2 is equal to V. This can be done by showing that every vector in V can be written as the sum of a vector in W1 and a vector in W2, and that this representation is unique.
Yes, V can be a direct sum of more than two subspaces. The definition of a direct sum applies to any number of subspaces, as long as the intersection of all the subspaces is only the zero vector and their sum is equal to V.
No, the subspaces W1 and W2 are not uniquely determined if V is a direct sum of them. The direct sum is unique, but the choice of subspaces W1 and W2 is not. There may be multiple combinations of subspaces that satisfy the conditions for a direct sum.
Understanding direct sums of subspaces is important in linear algebra and functional analysis. It has applications in fields such as physics, engineering, and computer science, where vector spaces and subspaces are used to model real-world systems. It is also used in solving systems of linear equations and in constructing orthogonal bases for vector spaces.