Question

Determine whether the given set of functions is linearly independent on the interval(-[infinity],[infinity]).

f₁(x) = 2, f₂(x) = cos²x, f₃(x) = sin²x

Answer 1

To determine whether the set of functions {f₁(x) = 2, f₂(x) = cos²x, f₃(x) = sin²x} is linearly independent, we can examine whether there exist constants c₁, c₂, and c₃, not all zero, such that:

\[ c₁f₁(x) + c₂f₂(x) + c₃f₃(x) = 0 \]

for all x in the given interval \((-∞, ∞)\). If such constants exist and not all of them are zero, then the functions are linearly dependent; otherwise, they are linearly independent.

Let's check the condition for linear independence:

\[ c₁(2) + c₂(\cos²x) + c₃(\sin²x) = 0 \]

For linear independence, the only solution should be \(c₁ = c₂ = c₃ = 0\).

Considering the constant term, we have \(c₁(2) = 0\), which implies \(c₁ = 0\). Now, the equation becomes:

\[ c₂(\cos²x) + c₃(\sin²x) = 0 \]

This equation holds only if \(c₂ = c₃ = 0\), as otherwise, the trigonometric terms can't cancel each other for all x.

Therefore, the set of functions {f₁(x) = 2, f₂(x) = cos²x, f₃(x) = sin²x} is linearly independent on the interval \((-∞, ∞)\).

Answer 2

The set of functions {2, cos²x, sin²x} is linearly dependent on the interval (-∞, ∞).

Explanation:

The functions cos²x and sin²x can be related using the trigonometric identity cos²x + sin²x = 1, which holds true for all x. This relationship demonstrates that one function can be expressed as a linear combination of the other two functions, making the set linearly dependent.

Specifically, if you subtract sin²x from both sides of the identity, you get cos²x = 1 - sin²x, or vice versa, indicating that either of these trigonometric functions can be written in terms of the other. Consequently, there exists a non-trivial combination (namely, multiplying one function by -1 and adding it to the other) that equals zero, confirming their linear dependence. Thus, in the interval (-∞, ∞), these functions cannot form a linearly independent set.

This dependency can be proven mathematically by showing that there exist constants (not all zero) such that the equation c₁ * 2 + c₂ * cos²x + c₃ * sin²x = 0 holds true for all x in the given interval. Using the identity mentioned earlier, we can manipulate this equation to demonstrate that it's satisfied by more than just the trivial solution where all constants are zero, confirming the linear dependence of these functions on the entire real number line.