In the context of vector spaces, if W1 and W2 are subspaces of a vector space V such that W1 ∩ W2 = {0} and dim(W1) + dim(W2) = dim(V) = n, then it is established that V is the direct sum of W1 and W2, denoted as V = W1 + W2. This conclusion is supported by the dimension formula, which states that dim(W1 + W2) + dim(W1 ∩ W2) = dim(W1) + dim(W2). Therefore, the conditions provided confirm that V is indeed the direct sum of the two subspaces.
PREREQUISITESStudents and professionals in mathematics, particularly those studying linear algebra, vector spaces, and their properties. This discussion is beneficial for anyone looking to deepen their understanding of direct sums and subspace relationships.
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?