please show steps 9. Use the information that the functions 1, x, x², ...,x" are linearly independent on the real line and the definition of linearindependence to prove directly that, for any constant r, the functions fo(x) = e TX, f, (x) = xe TX, f2(x) = x-e ™x,...%3Dfn(X) = x" e are linearly independent on the whole real line.i...If the functions f1, f2, f3, ., fn are linearly independent on the real line, then the identity c, f, (x) + C2f2(x) + ... + Cnfn (x) = 0%3Dholds on I only when (1)To prove that the functions fo(x) = e TX, f1(x) = xe "X, f2(x)= x²e x, .., fn(X) = x^e TX are linearly independent, thecorresponding identity is (2)What is the next step to prove the given functions are linearly independent?O A. Write the above sum in terms of the linear independent functions 1, x,x2x by isolating thecommon factor, xe rO B. Write the above sum in terms of the linear independent functions 1, x, x..., x" by isolating thecommon factor, x"erx.O C. Write the above sum in terms of the linear independent functions 1, x, x,uXby isolating thecommon factor, x²erx.(2)D. Write the above sum in terms of the linear independent functions 1, x, x, ...,x by isolating thecommon factor, e rx.Apply the above property to the identity. Thus,= 0.(3)Since the product of two terms is zero, either (3)Can e rX be zero?ON OO YesThus, (4),2is true, if and only ifBut the functions 1, x, x, .., x" are linearly independent. Therefore, (5)(6)the functions fo (x)= e rX, f,(x) = xe rX, f2(x) = x²e Tx,f,(x) = x"e rx are linearly independent on2X =As (7).%3Dthe whole real line.

Question

please show steps 9. Use the information that the functions 1, x, x², ...,x&#34; are linearly independent on the real line and the definition of linearindependence to prove directly that, for any constant r, the functions fo(x) = e TX, f, (x) = xe TX, f2(x) = x-e ™x,...%3Dfn(X) = x&#34; e are linearly independent on the whole real line.i...If the functions f1, f2, f3, ., fn are linearly independent on the real line, then the identity c, f, (x) + C2f2(x) + ... + Cnfn (x) = 0%3Dholds on I only when (1)To prove that the functions fo(x) = e TX, f1(x) = xe &#34;X, f2(x)= x²e x, .., fn(X) = x^e TX are linearly independent, thecorresponding identity is (2)What is the next step to prove the given functions are linearly independent?O A. Write the above sum in terms of the linear independent functions 1, x,x2x by isolating thecommon factor, xe rO B. Write the above sum in terms of the linear independent functions 1, x, x..., x&#34; by isolating thecommon factor, x&#34;erx.O C. Write the above sum in terms of the linear independent functions 1, x, x,uXby isolating thecommon factor, x²erx.(2)D. Write the above sum in terms of the linear independent functions 1, x, x, ...,x by isolating thecommon factor, e rx.Apply the above property to the identity. Thus,= 0.(3)Since the product of two terms is zero, either (3)Can e rX be zero?ON OO YesThus, (4),2is true, if and only ifBut the functions 1, x, x, .., x&#34; are linearly independent. Therefore, (5)(6)the functions fo (x)= e rX, f,(x) = xe rX, f2(x) = x²e Tx,f,(x) = x&#34;e rx are linearly independent on2X =As (7).%3Dthe whole real line.

Accepted Answer

We apply rule of linearly independent of sets