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The following question is what I want to solve:

Given that V = ${(a,b)}$ is a vector space, and addition is defined as $(a,b) + (c,d)$ = $(a + c - 1, b + d)$, and multiplication is defined as $r • (a,b)$ = $(ra - r + 1, rb)$.

Find a basis for V.

So far, I know that I have to build up a basis. I must start with a point, then add to it, until it spans V, and is linearly independent, but I don't know what to do from there.

Ian Murphy
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    This might sound silly, but did you start with a point? What point did you start with? Did you choose another point? Did it turn out to be a multiple of the other point? What went wrong with your intent? – davidlowryduda Mar 10 '15 at 16:18

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I suppose that all $a,b,r,..$are real numbers.

Note that the neutral element of the vector space is $(1,0)$.

Take the vector $v_1=(0,1)$, its span is $(-r+1,r)$ ($r$ is a scalar), so you can built a basis taking an other vector that is not in this span, and this vector can be $v_2=(0,-1)$. Note that his span is $(-r+1,-r)$ and the only common element with the span of $v_1$ is the neutral element.

Emilio Novati
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