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Which of the following sets of vectors are bases for R^2 ? a). {(0, 1), (1, 1)} b). {(1, 0), (0, 1), (1, 1)} c). {(1, 0), (−1, 0} d). {(1, 1), (1, −1)} e). {((1, 1), (2, 2)} f ). {(1, 2)}

MIT 18.06 Linear Algebra, Spring 2010
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so for this, the concept is, which of these vectors can serve as the basis for constructing the 2-d plane of real numbers...so imagine you have two vectors and you can multiply them by values and add them together...so you want to see if you can construct every point in the 2-d plane with just manipulating 2 vectors
so f is ruled out....you would only get vectors of varying magnitude
i think e can be ruled out for a similar concept: (2,2) is just (1,1) with twice the magnitude

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Other answers:

b cannot be a basis for R^2 since there are 3 vectors, so one must be a lincomb of the others i.e. (1,1) = 1(1,0) + 1(0,1)
c is not because one vector is a lincomb of the other (-1,0) = -1(1,0). Which leaves a and d, which are both bases in R^2 as neither contain lincombs.
The answer is (d). The dot product of the two vectors provided in (d) is 0, which means they are orthogonal and therefore form a basis for R^2. The following fail because: (a) : the vectors are not orthogonal to each other (b) : any basis for R^2 contains two vectors (c) : the two vectors lie on the same line (they are scalar multiples of each other) (e) : the two vectors lie on the same line (they are scalar multiples of each other) (f) : any basis for R^2 contains two vectors

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