Final answer:
A set of vectors in R^n that contains the zero vector is linearly dependent because the null vector can be expressed as a trivial linear combination of the other vectors in the set, using scalar multiplication by zero.
Step-by-step explanation:
If a set S={v1,...,vp} in R^n contains the zero vector, then the set is indeed linearly dependent. To understand this concept, we need to explore the definition of linear independence. A set of vectors is considered linearly independent if no vector in the set can be written as a linear combination of the others. In simple terms, this means you cannot multiply some vectors by certain scalars and add them together to get another vector in the set.
However, when the set includes the null vector, which is defined as a vector where all components are zero, the situation changes. The null vector can always be represented as a linear combination of other vectors in the set by simply multiplying those vectors by the scalar zero.
For example, 0 = 0*v1 + 0*v2 + ... + 0*vp, where 0 denotes the scalar zero and not the null vector. This means that the presence of the null vector allows us to write it as a trivial linear combination of other vectors, thus showing that the set is linearly dependent.
Since the null vector has no length and no direction, including it in any set of vectors in R^n instantly guarantees the set's linear dependence, because the null vector does not contribute any unique direction or magnitude to the set.