find all vectors spaces that have exactly one basis

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Today we explore a fascinating question in linear algebra: which vector spaces have exactly one basis? Recall that a basis is a linearly independent set that spans the entire vector space. Most vector spaces have infinitely many different bases, but we'll discover the rare exceptions where only one basis exists. Let's examine our first case: the zero vector space. This space contains only the zero vector. For a basis, we need a linearly independent set that spans the space. The empty set is linearly independent by definition, and its span is the zero vector, which equals our entire space. Therefore, the empty set is the unique basis for the zero vector space. Now consider one-dimensional vector spaces. If V has dimension 1, then V equals the span of some non-zero vector v. Any basis must have the form {cv} where c is a non-zero scalar. The crucial question is: how many non-zero scalars exist? In most fields, there are infinitely many, giving infinitely many bases. But in the field F-2 with only elements 0 and 1, there's only one non-zero scalar: c equals 1. For higher dimensions, we can always construct multiple bases. Starting with any basis B-1 containing vectors v-1, v-2, and so on, we can create a different basis B-2 by replacing v-1 with v-1 plus v-2. Since the dimension is greater than 1, v-2 is non-zero, making v-1 plus v-2 different from v-1. Both sets remain linearly independent and span the space, proving that higher-dimensional spaces have multiple bases. We have found the complete answer. There are exactly two types of vector spaces with exactly one basis: first, the zero vector space containing only the zero vector, which has the empty set as its unique basis; and second, one-dimensional vector spaces over the field F-2 with two elements. These are special because the zero space can only have the empty set as a basis, while F-2 spaces have only one non-zero scalar. All other vector spaces have multiple bases due to having multiple non-zero scalars or higher dimensions.