91Ó°ÊÓ

A linear combination involves expressing a vector as a sum of scaled versions of other vectors. Consider the vector \( \mathbf{x} = \begin{bmatrix} 2 \ -6 \end{bmatrix} \). To express \( \mathbf{x} \) in terms of a linear combination of basis vectors, we need to find coefficients \( c_1 \) and \( c_2 \) such that:

• \( \mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 \)

The vectors \( \mathbf{b}_1 = \begin{bmatrix} 3 \ -5 \end{bmatrix} \) and \( \mathbf{b}_2 = \begin{bmatrix} -4 \ 6 \end{bmatrix} \) form a basis for the space.
To find these coefficients, solve the equation \( B\mathbf{c} = \mathbf{x} \), where \( \mathbf{c} = \begin{bmatrix} c_1 \ c_2 \end{bmatrix} \). This involves working with the inverse matrix, but it essentially boils down to correctly scaling and summing the basis vectors to recreate \( \mathbf{x} \).

A basis matrix is formed from a set of basis vectors, which are linearly independent and span a vector space. For example, the basis \( \mathcal{B} \) given by:

• \( \mathbf{b}_1 = \begin{bmatrix} 3 \ -5 \end{bmatrix} \)
• \( \mathbf{b}_2 = \begin{bmatrix} -4 \ 6 \end{bmatrix} \)

can be organized into a basis matrix \( B \) as:

• \( B = \begin{bmatrix} 3 & -4 \ -5 & 6 \end{bmatrix} \)

The significance of a basis matrix is its ability to transform and represent vectors in a new coordinate system defined by the basis. By using the inverse of this matrix, we can convert back and forth between the standard coordinate system and this new system. This is critical in understanding vector spaces and transformations, providing us a clear method to express vectors in different bases.

The determinant of a matrix is an essential number that indicates whether a matrix is invertible. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \).
In our example, the determinant of the basis matrix \( B \) is:

• \( \text{det}(B) = 3 \times 6 - (-4) \times (-5) = 18 - 20 = -2 \)

Since this value is non-zero, \( B \) is invertible, meaning it has an inverse matrix. The non-zero determinant is crucial because it ensures that the basis matrix can be used to find unique solutions when translating vectors between different bases.
It also provides information on the stability of a matrix's transformations. If the determinant were zero, the matrix would be singular, unable to map back to its original form after transformation.

Invertibility is a key feature of a matrix that allows it to have a unique inverse. For a matrix to be invertible, it must have a non-zero determinant.
In our exercise, we calculated the determinant of matrix \( B \) to be \(-2\), so we confirmed that \( B \) is invertible. This inversible attribute enables the computation of \( B^{-1} \), the inverse of \( B \).

• The inversion involves switching the positions of terms and altering signs: \( B^{-1} = \frac{1}{-2} \begin{bmatrix} 6 & 4 \ 5 & 3 \end{bmatrix} = \begin{bmatrix} -3 & -2 \ -2.5 & -1.5 \end{bmatrix} \)

Invertibility is significant because it ensures that a system of linear equations represented by the matrix can be uniquely solved. Every input vector has a unique corresponding vector in the basis representation, and vice-versa. Being invertible makes the matrix an effective tool for various transformations and real-world applications.