Let fx x and gx x Show that f and g are linearly independe

Let f(x) = x and g(x) = |x|. Show that f and g are linearly independent on C[-1,1] and linearly dependent on C[0,1].

Solution

The given functions f(x) = x and g(x) = |x| are linearly independent since otherwise there would be nonzero constants C1 and C2 such that

C1*X + c2*|X| = 0

For all X, C[-1,1]

Let X = -1 Then (C1)(-1) + (C2)(|-1|) = 0

And, let X = 1 Then (C1)(1) + (C2)(|1|) = 0

These equations will only satisfy if C1 = C2 = 0

Therefore Given two functions are linearly independent for C[-1,1]

Same for C[0,1]

Let X = 0 Then (C1)(0) + (C2)(|0|) = 0

Here, C1 = C2 0

And, for X = 1 Then (C1)(1) + (C2)(|1|) = 0

These equations will satisfy if C1 = C2 = 0

Therefore Given two functions are linearly dependent for C[0,1]