If u is a vector and k is a scalar such that ku = 0, then it must be true that k = 0.
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If u is a nonzero vector and k is a scalar such that ku = 0, then it must be true that k = 0.
To see why this is true, suppose that u is a nonzero vector, and that ku = 0 for some scalar k. Then either k is zero, or u is a zero vector. But if u is nonzero, then it cannot be the zero vector, so we must have k = 0.
To put it another way, if k is nonzero, then ku is a nonzero scalar multiple of u, which means that ku cannot be equal to zero unless u is the zero vector. Therefore, if ku = 0 and u is nonzero, then k must be zero.
So the statement is true: if u is a nonzero vector and k is a scalar such that ku=0, then it must be true that k=0.
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