Final answer:
To show that the set of functions {e²ˣ, xe²ˣ, (x²+1)e²ˣ} is linearly independent, we need to show that no non-trivial linear combination of these functions can equal the zero function, except when the coefficients are all zero.
Step-by-step explanation:
To show that the set of functions {e²ˣ, xe²ˣ, (x²+1)e²ˣ} is linearly independent, we need to show that no non-trivial linear combination of these functions can equal the zero function, except when the coefficients are all zero.
Let's assume that there exist constants a, b, and c (not all zero) such that ae²ˣ + bxe²ˣ + c(x²+1)e²ˣ = 0 for all values of x. We can rewrite this equation as e²ˣ(a + bx + c(x²+1)) = 0.
Since e²ˣ is never zero for any value of x, we can divide both sides of the equation by e²ˣ to get a + bx + c(x²+1) = 0. However, this is a quadratic equation and its highest power of x is 2. A quadratic equation can't be zero for all values of x unless all of its coefficients are zero. Therefore, a = b = c = 0 and the set of functions is linearly independent.