Idea:-
Dear Students you know that in differential equations we determine the functions that satisfy the differential equations. Some times the functions are linearly dependent and some times they are linearly independent. In each of the following questions, please determine whether the given functions are linearly independent or dependent for the given intervals. Justify your answer.
(1) f (x) = Absolute (x) and f (x) = x on ( 0 , 10 )
Interval is (0,10). Note that this is NOT [0,10] i.e. 0 and 10 are itself not included. Anyhow, even if we include 0 and 10 in this part, then our answer will still be the same.
According to absolute value function y=|x| means y =-x when x<0 and y=x when x>=0, since in our case x>0, therefore, we will have y=x. i.e. f(x)= Absolute(x) equals x on interval (0,10). Now, both functions are same, therefore, they are linearly DEPENDENT.
(2) f (x) = Absolute (x) and f (x) = x on ( -10 , 0 )
Here, our interval is from greater than -10 and less than 0.
For negative values of x, absolute(x) becomes –x, which is not same as the second function. We have two different functions y = -x and y=x, One function is CONSTANT multiple of another; therefore, they are linearly DEPENDENT.
(1) f (x) = Absolute (x) and f (x) = x on ( 0 , 10 )
Interval is (0,10). Note that this is NOT [0,10] i.e. 0 and 10 are itself not included. Anyhow, even if we include 0 and 10 in this part, then our answer will still be the same.
According to absolute value function y=|x| means y =-x when x<0 and y=x when x>=0, since in our case x>0, therefore, we will have y=x. i.e. f(x)= Absolute(x) equals x on interval (0,10). Now, both functions are same, therefore, they are linearly DEPENDENT.
(2) f (x) = Absolute (x) and f (x) = x on ( -10 , 0 )
Here, our interval is from greater than -10 and less than 0.
For negative values of x, absolute(x) becomes –x, which is not same as the second function. We have two different functions y = -x and y=x, One function is CONSTANT multiple of another; therefore, they are linearly DEPENDENT.
