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I understand that the following is a linear combination of vectors $y_{1},..., y_{n}$:

$a_{1}y_{1} + a_{2}y_{2} + ... + a_{n}y_{n}$

My doubt is whether, the following which has an additional constant term $a_{0}$, can also be called a linear combination of vectors $y_{1},..., y_{n}$:

$a_{0} + a_{1}y_{1} + a_{2}y_{2} + ... + a_{n}y_{n}$

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I assume your $a_i$s are scalars and your $y_j$s are vectors. What does it even mean to add a scalar to a vector? – Chris Eagle Aug 23 '11 at 10:20
up vote 6 down vote accepted

No, it isn't. In fact, it doesn't make any sense, since $1$ -appearing in the first term as $a_0\cdot 1$, is not supposed to be a vector of your vector space, in general.

For instance, assume your vector space is $\mathbb{R}^2$: can you add $1$ to $(5,-3)$?

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For some reason when combined with the OP's nickname and acceptance of the answer, I remembered my psychology while I was writing my thesis. :) – user13838 Aug 23 '11 at 13:28

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