The vectors (1,a) and (1,b) in R² are linearly independent if and only if a ≠ b. This conclusion arises from the requirement that the only solution to the equation x(1,a) + y(1,b) = (0,0) is the trivial solution where both x and y are zero. If a equals b, the vectors become scalar multiples of each other, leading to linear dependence. Therefore, the condition for linear independence is strictly tied to the distinctness of the scalar values a and b.
PREREQUISITESStudents and educators in mathematics, particularly those focusing on linear algebra, as well as anyone seeking to deepen their understanding of vector relationships in R².
Good hint!Deveno said:what does it mean for two vectors to be linearly independent?
(hint: the answer is NOT:
there exist c1,c2 such that c1v1+c2v2 = 0).