Final answer:
To show that y = ex and y = e2x are linearly independent, we need to show that no linear combination of the two equations can equal zero for all values of x.
Step-by-step explanation:
To show that y = ex and y = e2x are linearly independent, we need to show that no linear combination of the two equations can equal zero for all values of x.
Let's assume that a and b are constants, and we have a linear combination of the two equations: ay = ex + be2x.
If we choose x = 0, then the equation becomes ay = 1 + b. Since the equation must hold true for all values of x, it means that a and b must be equal to 0 in order for the equation to equal 0. But if a and b are both 0, then the equation is just 0 = 0, which is not a valid linear combination.
Therefore, y = ex and y = e2x are linearly independent.