If H is a pdimensional subspace of Rn and G is a pdimensiona

If H is a p-dimensional subspace of R^(n) and G is a p-dimensional subspace of R^(n) contained in H, then G = H

Solution

Since G is p-dimensional, it has a basis consisting of p linearly independent vectors. Since G is contained in H, these vectors belong to H. Since they are linearly independent and H is p-dimensional, they form a basis for H. Thus G and H both have these p vectors as a basis and so they must be equal.

The problem states that G is contained in H. This means that GH or in other words, for any xG, we also have xH