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A linear combination involves expressing a vector (or matrix) as a sum of other vectors, each multiplied by a constant. This concept is crucial when dealing with basis matrices because it allows us to describe any matrix in the space using these basis elements. For instance, in the context of our exercise, we are asked to express matrix \(A\) as a linear combination of the basis matrices \(B_1, B_2, B_3,\) and \(B_4\). This means we find constants \(c_1, c_2, c_3,\) and \(c_4\) such that:\[A = c_1B_1 + c_2B_2 + c_3B_3 + c_4B_4\]The coefficients \(c_1, c_2, c_3,\) and \(c_4\) tell us how much of each basis matrix we need to use to reconstruct \(A\). Solving for these coefficients involves equating the sum of these weighted basis matrices to the original matrix \(A\), element by element.

Basis matrices are fundamental to understanding how spaces of matrices are constructed. A basis provides a set of matrices that are linearly independent and can combine linearly to describe any matrix in the space. In our exercise, the basis \(\mathcal{B}\) for \(M_{22}\) consists of four specific matrices:

• \(B_1 = \begin{bmatrix} 1 & 0 \ 0 & 0 \end{bmatrix}\)
• \(B_2 = \begin{bmatrix} 1 & 1 \ 0 & 0 \end{bmatrix}\)
• \(B_3 = \begin{bmatrix} 1 & 1 \ 1 & 0 \end{bmatrix}\)
• \(B_4 = \begin{bmatrix} 1 & 1 \ 1 & 1 \end{bmatrix}\)

These matrices serve as building blocks for any other matrix within this space of 2x2 matrices. If you can express any 2x2 matrix as a linear combination of these four matrices, they form a basis. They effectively "span" the space, meaning their linear combinations can cover all possible 2x2 matrices.

Matrix representation is the written portrayal of a matrix by expressing it in terms of basis matrices and their corresponding coefficients. In this exercise, matrix \(A\) is represented as:\[A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\]Using the basis \(\mathcal{B}\), we rewrite \(A\) in a form that makes its relationship to these basis matrices clear. The matrix's representation in this basis consists of the coefficients \([-1, -1, -1, 4]\), showing how much of \(B_1, B_2, B_3,\) and \(B_4\) is combined to produce \(A\). This representation provides a deeper understanding of how \(A\) changes when considering different bases.

The notation \(M_{22}\) refers to the space of all 2x2 matrices. These matrices have two rows and two columns. Such a space includes all possible combinations of four numbers arranged into 2x2 matrices. Here's how matrix \(A\) fits into \(M_{22}\):\[A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}\]This space is important because operations like matrix addition and scalar multiplication are well defined, and this space is vector-like in structure. Knowing the basis for \(M_{22}\) allows us to describe any of its matrices through linear combinations. This forms a conceptual bridge between abstract algebraic structures and concrete matrix operations, making complex matrix calculations manageable.