Answer and Step-by-step explanation:
Solution:
(a) The function. f1(x) = ln x, f2(x) = ln x^5 is linearly independent or not.
According to “Linear independent theorem: if two functions f1and f2 are linearly dependent if one can be represent as the constant multiple of other.
f1 = n. f2
Consider given function:
f1(x) = lnx f2(x) = lnx5
(b) Set of function f1(x) = x^n, f2(x) = x^n + 2, is linearly independent or not on the interval [-∞, ∞].
According to “Linear independent theorem: if two functions f1and f2 are linearly dependent if one can be represent as the constant multiple of other.
f1 = n. f2
Consider the given function:
f1(x) = xn f2(x) = xn+2
(c) f1(x) = x, f2(x) = x + 8, (−[infinity], [infinity])
Function is not linearly independent.
Linear independent theorem states that two functions f1 and f2 are linearly dependent if one can represent as the constant multiple of other.
F1= n.f2
(d) f1(x) = cos(x + π), f2(x) = sin x. [-∞, ∞]
According to “Linear independent theorem: if two functions f1and f2 are linearly dependent if one can be represent as the constant multiple of other.
f1 = n. f2
The given function:
f1(x) = cos(x + π)f2(x)
