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I have heard this used, within the context of results between the same up to a scalar, but I'm not sure of its meaning.

Can anyone provide an explanation and example in as simple terms as possible?

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1 Answer

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To say that two things are the same up to a scalar multiple means that either of them is a scalar multiple of the other, and they are therefore considered equivalent.

An example is linear dependence among vectors. Suppose $4\vec a + 2\vec b-9\vec c = \vec 0$, so that $(4,2,-9)$ is a linear dependence among the vectors $\vec a,\vec b,\vec c$. Then every nonzero scalar multiple of $(4,2,-9)$ is also a linear dependence among the vectors $\vec a,\vec b,\vec c$, and is essentially the same linear dependence. Thus one might say that up to a scalar multiple, $(4,2,-9)$ is the only linear dependence among these vectors.

For further examples see this Wikipedia article.

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