Final answer:
The set S1 = {a, b, c, d, x} is linearly independent because x is not in the span of the original linearly independent set S, therefore no vector in the extended set S1 can be expressed as a linear combination of the others.
Step-by-step explanation:
If a set S = {a, b, c, d} is a linearly independent set in a vector space V, and x is a vector in V that is not in the span of S, then the expanded set S1 = {a, b, c, d, x} is also linearly independent.
By definition, a set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. Since x is not in the span of S, this means that there is no combination of vectors a, b, c, and d that can be combined to produce x. Consequently, adding x to the set S does not introduce any linear dependencies, and thus the set S1 remains linearly independent.
A concrete example of this would be if vectors a, b, c, and d represent four non-coplanar vectors in three-dimensional space, and x is a vector that doesn't lie on the space spanned by a, b, c, and d. Including x to the set would not change the fact that none of the vectors in the set can be built from a linear combination of the others.