Showing V is the direct sum of W1 and W2

[La_Lune]

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?

 

[jbunniii]

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?

Yes, because $\dim(W1+W2) + \dim(W1 \cap W2) = \dim(W1) + \dim(W2)$.