check if functions are linearly independent
How do you verify that the given functions are linearly independent?
Let f and g be differentiable on [a,b].
If Wronskian W(f,g)(t0) is nonzero for some t0 in [a,b] then f and g are linearly independent on [a,b].
If f and g are linearly dependent then the Wronskian is zero for all t in [a,b].28 juil. 2023If a consistent system has exactly one solution, it is independent .
If a consistent system has an infinite number of solutions, it is dependent .
When you graph the equations, both equations represent the same line.
How do you know if a function is linearly independent?
One more definition: Two functions y 1 and y 2 are said to be linearly independent if neither function is a constant multiple of the other.
For example, the functions y 1 = x 3 and y 2 = 5 x 3 are not linearly independent (they're linearly dependent), since y 2 is clearly a constant multiple of y 1.
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Hint: Show that these functions are linearly independent if and only if 1, x, x2, , xm−1 are linearly independent Solution: We will use two methods to arrive at |
