In every vector space the vectors (−1)u and −u are the same
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In every vector space, the vectors (-1)u and -u are equivalent, or the same. This is because (-1)u is defined as the scalar product of the vector u and the scalar -1, which results in the opposite direction of u with the same magnitude.
On the other hand, the vector -u is the additive inverse of u, meaning that when added to u, the resulting vector will be the zero vector. This is because the sum of a vector and its additive inverse is always zero, which is the identity element of the vector space with respect to vector addition.
Hence, since (-1)u and -u produce the same opposite direction of vector u, they can be considered equivalent or the same vector in every vector space.
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