# Thread: dimensional subspaces and direct sums

1. ## dimensional subspaces and direct sums

1. Prove that if $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space V, then the subspace $W_1$+ $W_2$ is finite-dimensional and dim( $W_1$+ $W_2$)=dim( $W_1$)+dim( $W_2$)-dim( $W_1$n $W_2$).

2. Let $W_1$ and $W_2$ be finite-dimensional subspaces of a vector space V, and let V= $W_1$+ $W_2$. deduce that V is the direct sum of $W_1$ and $W_2$ iff dim(V)=dim( $W_1$)+dim( $W_2$)

2. Originally Posted by studentmath92
1. Prove that if $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space V, then the subspace $W_1$+ $W_2$ is finite-dimensional and dim( $W_1$+ $W_2$)=dim( $W_1$)+dim( $W_2$)-dim( $W_1$n $W_2$).

2. Let $W_1$ and $W_2$ be finite-dimensional subspaces of a vector space V, and let V= $W_1$+ $W_2$. deduce that V is the direct sum of $W_1$ and $W_2$ iff dim(V)=dim( $W_1$)+dim( $W_2$)
These are results that should be available in most of the std texts. Where are you getting stuck?