Thread: dimensional subspaces and direct sums

1. dimensional subspaces and direct sums

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

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

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

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