Linear algebra which of the following are bases Which of th

Linear algebra - which of the following are bases?

Which of the following are \"bases\" in the way described? (2, 3) and (-1, 1) generate R^2 over the real numbers. 2^-2, 2^-1, 2^0, 2^1, 2^2, ... generate real numbers by using their binary representation over F^2. 1 and Squareroot 2 generate the algebraic numbers over the rationals.

Solution

Let us consider any (x,y) in R^2 where x and y are real

If possible let (x,y) = a(2,3)+b(-1,1)

x = 2a-b

y = 3a+b

Adding 5a = x+y or a = (x+y)/5 and b = (3x-y)/5

Thus for every x and y scalars a and b exist.

Hence (2,3) (-1,1) generate R^2

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2) Any number in binary represented as linear combination of powers of 2.

Hence a basis.

3) Cannot form basis as rt 2 is irrational and hence combination of rt 2 will not be rational