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Prove that K^n satisfies axioms of vector spaces

Let K be a field and n greater than or equal to 0. Prove that K^n satisfies the following axioms of vector spaces:

1) (a + b) + c = a + (b + c)

2) There is a number 0 in S such that a + 0 = 0 + a = a for all a in S.

I don't understand what to do as I don't really understand what K^n is and how I'd go about proving these statements.

Thanks.
Original post by Benniboi1
Let K be a field and n greater than or equal to 0. Prove that K^n satisfies the following axioms of vector spaces:

1) (a + b) + c = a + (b + c)

2) There is a number 0 in S such that a + 0 = 0 + a = a for all a in S.

I don't understand what to do as I don't really understand what K^n is and how I'd go about proving these statements.

Thanks.


You can think of K^n as an n-tuple whose components are elements of K.

Kn={(a1,a2,,an)aiK for i=1,,n} \mathbb{K}^n=\{(a_1,a_2,\ldots,a_n )| a_i\in\mathbb{K}\text{ for }i=1,\ldots, n\}

n=0 might be a problem though, unless it's just the zero element.
(edited 10 years ago)

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