If L=(a1)(s1) + (a2)(s2) + ... + (an)(sn) where a1, a2, ... ,an are scalar and s1, s2, ... sn are vectors, then L is a linear combination of s1, s2, ... , sn.

In English, that means that if you have some vectors, which are just n-tuples (pairs of numbers, or triplets, etc.), and you take a combination of them, for example one of the first, three of the second, and none of third, then that combination is called a linear combination of those three vectors. The reason it's called "linear" is because you are just taking a multiple of them, you're not multiplying them by anyting nasty, or squaring them, or anyting like that.

This term is used a lot in linear algebra.

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