91Ó°ÊÓ

Understanding linear independence is fundamental when dealing with vectors and spaces. In a simplest form, a set of vectors in a vector space is said to be linearly independent if no vector in the set can be written as a linear combination of the others. Here, we apply this concept to the functions forming the set \( B = \{ \sin(x), \cos(x), \sin(x)\cos(x), \sin^2(x), \cos^2(x) \} \). These functions form a basis if they are linearly independent.
Assume you have a linear combination of these functions equating to zero: \[ a_1 \sin(x) + a_2 \cos(x) + a_3 \sin(x)\cos(x) + a_4 \sin^2(x) + a_5 \cos^2(x) = 0. \] For these functions to be independent, the only solution is if all coefficients \(a_1, a_2, a_3, a_4,\) and \(a_5\) are zero. If this holds true, none of the functions in \( B \) can be expressed as a combination of the others, proving linear independence.

Once we've confirmed linear independence, the next step is ensuring the set of functions forms a basis by spanning the desired vector space. For \( B \) to be a basis of \( V \), it needs to both be linearly independent and span \( V \).
Since \( V \) is defined as the span of \( B \), a linear space consisting of all linear combinations of the functions in \( B \), \( B \) already spans \( V \). Linear independence combined with this spanning property confirms that \( B \) indeed constitutes a basis for \( V \). In essence, every function in \( V \) can be expressed uniquely as a linear combination of the basis elements.

To study transformations, we use a transformation matrix, which represents the effect of applying a linear operator under a specific basis. For this exercise, the transformation matrix \( M_{\mathcal{B}}(F) \) corresponds to the endomorphism \( F \) that maps functions to their derivatives. Here's how it's done:

• Compute the derivative of each basis function.
• Express these derivatives in terms of the original basis functions.
• Each derivative creates a column in the transformation matrix, reflecting its coordinates relative to the basis \( B \).

Through this methodical approach, the transformation matrix \( M_{\mathcal{B}}(F) \) is constructed, encapsulating the entire action of \( F \) across the space.

The kernel and image are key concepts when examining linear transformations. The kernel consists of all vectors (or functions, in this case) that are mapped to zero by the transformation. For the derivative operator \( F \), these are constant functions, but specifically within \( V \), \( \operatorname{Ker}(F) \) is spanned by invariants under differentiation: \( \{ \sin^2(x), \cos^2(x) \} \).
The image of \( F \), on the other hand, consists of functions that are the result of differentiating basis elements of \( V \). What's distinct in this exercise is how the non-zero columns of the transformation matrix \( M_{\mathcal{B}}(F) \) align with \( \sin(x), \cos(x), \text{and } \sin(x)\cos(x) \). This reveals the image is spanned by these functions, symbolizing the scope of \( F \) across the vector space.