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Linear combinations are essential in understanding spans in vector spaces. A linear combination of vectors occurs when vectors are multiplied by scalars and then added together. This can be simply imagined as forming new vectors from known ones by stretching, shrinking, or flipping them based on the scalar values. For any vector set \( S = \{ \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_k \} \), its span consists of all possible vectors that can be formed by linear combinations of vectors from \( S \). Hence, if we say a vector \( \mathbf{v} \) is in \( \operatorname{span}(S) \), it means \( \mathbf{v} = c_1\mathbf{u}_1 + c_2\mathbf{u}_2 + \ldots + c_k\mathbf{u}_k \), where \( c_1, c_2, \ldots, c_k \) are scalars.
The beauty of linear combinations is that they describe how any vector within a span relates to its generating set. Each scalar is a possible weight, determining the contribution of each vector in \( S \) towards forming \( \mathbf{v} \). By understanding this process, the concept of spans becomes more intuitive.

Understanding subsets like \( S \) and \( T \) in vector spaces is a foundation of comparing spans. Given \( S \) is a subset of \( T \), it follows that \( T \) contains more or the same amount of elements as \( S \). This relationship implies that any span generated by \( S \) is inherently contained within the span of \( T \). This is because \( T \) can form every vector that \( S \) can form and potentially more due to the extra vectors.
When we say \( \operatorname{span}(S) \subseteq \operatorname{span}(T) \), it indicates that every linear combination resultant from vectors in \( S \) can also be represented with vectors in \( T \). For example, by setting additional vectors' coefficients in \( T \) to zero, any vector from \( S \) is reproducible. In this manner, the concept of a subset directly influences and limits the scope of linear combinations within a vector space.

Exploring \( n \)-dimensional spaces aids in grasping the dimensionality and complexity of spans. In the context of linear algebra, an \( n \)-dimensional space, such as \( \mathbb{R}^n \), represents a space where vectors have \( n \) components. When dealing with spans in \( \mathbb{R}^n \), if \( \operatorname{span}(S) = \mathbb{R}^n \), it implies that the span can form any vector in that space, meaning \( S \) spans the entire space.
Therefore, if \( \operatorname{span}(S) \) could cover \( \mathbb{R}^n \), adding more vectors to form \( T \) won't change its ability to span. Thus, given \( \operatorname{span}(S) = \mathbb{R}^n \) and knowing \( \operatorname{span}(S) \subseteq \operatorname{span}(T) \), it follows that \( \operatorname{span}(T) \) must span \( \mathbb{R}^n \) too.
This principle showcases the power and coverage of spans within \( n \)-dimensional spaces, providing a clearer understanding of how comprehensive a set of vectors is in defining an entire space.