Determine whether the statement is True or False. Justify your answer:If u is a vector in a vector space V, then (−1)u is the same as the negative of u.
The statement is true
The statement is true.
To justify this, we can use the properties of vector spaces. The scalar multiplication property states that for any vector u in a vector space V and any scalar c, the scalar multiple c*u is also a vector in V.
In this case, we are considering the multiplication of the vector u by the scalar -1, which is denoted as (-1)u. According to the scalar multiplication property, (-1)u is a vector in V.
Now, let’s consider the negative of u, denoted as -u. By definition, the negative of a vector u is a vector that, when added to u, gives the zero vector. In other words, u + (-u) = 0.
Since (-1)u is a vector in V and -u is also a vector in V, in order to justify that (-1)u is the same as -u, we need to show that (-1)u + u = 0.
Using the distributive property of scalar multiplication, we have:
(-1)u + u = (-1 + 1)u = 0u = 0.
So, (-1)u + u indeed equals the zero vector, which implies that (-1)u is the same as -u.
Therefore, the statement “If u is a vector in a vector space V, then (-1)u is the same as the negative of u” is true.
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