Answer :
In this problem, we are given two functions f(x) = sin(x) and g(x) = cos(x), and we need to perform several tasks related to their linear independence and span.
To show that f and g are linearly independent, we can use the Wronskian. The Wronskian of f and g is given by W(f, g) = f'(x)g(x) - f(x)g'(x). By calculating the Wronskian and showing that it is non-zero for all x, we can conclude that f and g are linearly independent.
For the functions f₁(x) = sin(x + a) and g₁(x) = cos(x + a), we need to show that they belong to the span of W, i.e., they can be expressed as linear combinations of f and g. By manipulating the expressions of f₁ and g₁ using trigonometric identities and properties, we can rewrite them in terms of f and g, demonstrating that f₁ and g₁ can be spanned by W.
To show that {f₁, g₁} is linearly independent, we need to prove that no non-trivial linear combination of f₁ and g₁ can equal the zero function. By assuming that there exist coefficients not all zero, we can construct a linear combination and show that it does not vanish, implying linear independence.
By performing these steps, we establish the linear independence of f and g using the Wronskian, demonstrate that f₁ and g₁ belong to the span of W, and prove the linear independence of {f₁, g₁}.
Learn more about Wronskian here
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