91影视

Let V be a vector space of dimension 鈥榤鈥�

Let us consider a set G consisting of 鈥榤鈥� vectors such that G is a finite generating set for V. Then by theorem, 鈥淚f a vector space V is generated by a finite set S , then some subset of S is a basis for V and hence, V has a finite basis鈥�, some subset H of G is a basis for V. Now by theorem, 鈥淚f V be a vector space having a finite basis, then every basis for V contains the same number of vectors鈥�. This implies that the subset H of G contains exactly 鈥榤鈥� vectors.

Since a subset H of G contains 鈥榤鈥� vectors then G must contains at least 鈥榤鈥� vectors.

Moreover, if G contains exactly 鈥榤鈥� vectors then we must have H=G, so that G is a basis for the vector space V.

Since the generating set G of vector space contains at least 鈥榤鈥� vectors, therefore, we need at least 鈥榤鈥� vectors to span a space of dimension 鈥榤鈥�. If G contains exactly 鈥榤鈥� vectors then we must have H=G, so that G is a basis for the vector space V.