22. Dimension of partial spaces

Let T denote a partial space of the n-dimensional vector space V. Since there cannot be in T more linearly independent vectors than in V, there must exist a number r £ n with the property that there is in T a system of r linearly independent vectors, while r + 1 vectors of T are linearly dependent. Hence T is a vector space of dimension r; every system of r linearly independent vectors of T forms a base. By the way, if r = n, then T coincides with the entire space V, because T contains then n linearly independent vectors, that is, a base of V.

In particular, let T be the partial space of V of the vectors

a₁, ··· , a_m (22.1)

and determine the dimension of T. For this purpose, select from a₁, ··· , a_m the largest possible system of linearly independent vectors.

Let such a system contain r vectors. You find that r is the dimension of T. This is immediately clear if r = m, because all vectors of T can be linearly composed from a₁, ··· , a_m due to their linear independence in one way; hence the a₁, ··· , a_m form a base of T. If r < m, since apparently the sequence of the vectors (22.1) is indifferent, you can assume, without restricting generality, that the a₁, ··· , a_m form the largest possible partial system of linearly independent vectors. If you add a vector a_s to the set a₁, ··· , a_m , you have r + 1 vectors between which there must exist one non-trivial relation

g₁a₁ + ··· + g_ra_r + g_sa_s = 0. (22.2)

Due to the linear independence of the a₁, ··· , a_m , you have here g_s ¹ 0, whence (22.2) yields a represenation of a_s as a linear combination of the a₁, ··· , a_r. Now you can represent every vector of T as a linear combination of a₁,···, a_m. In such a presentation, you can again express everyone of the vectors a_r+1,···,a_m in terms of a₁, ··· , a_r. Hence every vector of T can be represented as a linear combination of a₁, ··· , a_r alone. It follows from the linear independence of a₁, ··· , a_r that this representation is unique. Hence a₁, ··· , a_r form a base of T and T has the dimension r.

Theorem 11

The dimension of T equals the largest number of linearly independent vectors generating T.

In the sequel, the following, obvious observation will become useful: If you add to the vectors a₁, ··· , a_m, generating T, one vector a, which is linearly dependent on a₁, ··· , a_m, then the vector space, generated by a,a₁,···,a_m coincides with T. In fact, you only add to T a vector, which is already contained in T and which cannot be enlarged by it. In particular, there are among the vectors a,a₁,···,a_m just as many linearly independent vectors as in a₁,···,a_m.

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