91Ó°ÊÓ

In mathematics, a linear combination refers to an expression constructed from a set of terms by multiplying each term by a constant and adding the results. For the cosine space \( \mathbf{F}_{3}\), our linear combination involves trigonometric functions:

• \( \cos x \)
• \( \cos 2x \)
• \( \cos 3x \)

This creates the function: \[ y(x) = A \cos x + B \cos 2x + C \cos 3x \]Here, \(A\), \(B\), and \(C\) are the coefficients that allow us to adjust each term's contribution to the overall function. By substituting different values for \(A\), \(B\), and \(C\), we obtain different instances of functions within this cosine space.

A basis in a subspace is a set of vectors that are both linearly independent and span the entire subspace. This means that any vector in the subspace can be expressed as a linear combination of the basis vectors. For our function, we are tasked to find a basis that meets the condition \(y(0)=0\).
When we solve the condition, we find that \(A + B + C = 0\). By expressing one variable in terms of others, like \(C = -A - B\), we adjust the terms and redefine our function:\[ y(x) = A (\cos x - \cos 3x) + B (\cos 2x - \cos 3x) \]The above expression implies the basis for the subspace is:

• \(\cos x - \cos 3x\)
• \(\cos 2x - \cos 3x\)

These two functions can be used to form any other function within this subspace, hence they constitute the basis.

Linear independence is an attribute of a set of vectors wherein no vector in the set is written as a linear combination of others. To verify the linear independence of our basis vectors:
Assume a linear combination results in zero:\[ p(\cos x - \cos 3x) + q(\cos 2x - \cos 3x) = 0 \]Expanding gives:\[ p \cos x + q \cos 2x - (p+q) \cos 3x = 0 \]For the linear independence, each coefficient must individually be zero, leading to the conditions:

• \(p \cos x = 0\)
• \(q \cos 2x = 0\)
• \(-(p+q) \cos 3x = 0\)

Thus, \(p = 0\) and \(q = 0\), confirming that the vectors are linearly independent.

In trigonometric functions like the ones in cosine space \(\mathbf{F}_{3}\), coefficients play the crucial role of determining the scaling factor of each trigonometric term within the function. In the function:\[ y(x) = A \cos x + B \cos 2x + C \cos 3x \]The coefficients \(A\), \(B\), and \(C\) can be altered to give different amplitudes to \( \cos x \), \( \cos 2x \), and \( \cos 3x \) respectively.
When solving for the subspace where \(y(0) = 0\), these coefficients are crucial as they define the relationships between the cosine components. The constraint \(A + B + C = 0\) directly ties into how these trigonometric functions relate in the subspace, thus, affecting the entire structure and behavior of the function.