91Ó°ÊÓ

A linear combination involves expressing a vector as a sum of other vectors, each multiplied by a scalar. In our problem, we aim to express the vector \(\boldsymbol{c}\) as a linear combination of vectors \(\boldsymbol{a}\) and \(\boldsymbol{b}\).
This concept is fundamental in vector mathematics because it helps define vector spaces and enables various operations and transformations.

Here’s a simple breakdown:

• Vector \(\boldsymbol{c}\) can be broken down into components of \(\boldsymbol{a}\) and \(\boldsymbol{b}\).
• The scalars \(s\) and \(t\) are what give the necessary weights for this combination.

This is useful in solving many real-world problems, such as navigation and physics, where vectors represent directions and magnitudes.

In the process of finding a linear combination of vectors, we set up a system of linear equations. This system arises from equating the components of the vectors involved. For the case of vectors \(\boldsymbol{a}\), \(\boldsymbol{b}\), and \(\boldsymbol{c}\), we develop two equations based on their individual components:

• \(c_1 = s a_1 + t b_1\)
• \(c_2 = s a_2 + t b_2\)

The above equations form a linear system that we need to solve to find the values of \(s\) and \(t\).

Thinking about it, solving such a system is like untangling a web where each equation represents a line. The solution to the system gives the point at which these "lines" intersect, which corresponds to the scalars needed to express \(\boldsymbol{c}\) as a combination of \(\boldsymbol{a}\) and \(\boldsymbol{b}\).

Matrix inversion is a process used to find the unique solution to a system of linear equations when the matrix is invertible. An invertible matrix, known as a non-singular matrix, can have its inverse calculated.

Once we set up the problem like the following matrix equation:
\[\begin{pmatrix} a_1 & b_1 \ a_2 & b_2 \end{pmatrix}\begin{pmatrix} s \ t \end{pmatrix} = \begin{pmatrix} c_1 \ c_2 \end{pmatrix} \]The goal is to solve for the matrix \(\begin{pmatrix} s \ t \end{pmatrix}\).
We achieve this by multiplying both sides by the inverse of the coefficient matrix, granted the determinant is non-zero.

This inversion is essential as it transforms the multiplication into a straightforward solution calculation. It provides a neat solution to simultaneous equations by reducing them into simpler single-step calculations.

The determinant is a special number calculated from a square matrix. It is crucial as it determines whether a matrix is invertible or not. For our two-dimensional vectors, the matrix determinant must be calculated as follows:
\[\text{det}(\mathbf{A}) = a_1b_2 - a_2b_1\]

• If the determinant is zero, the matrix is singular, meaning there are no unique solutions.
• Non-zero determinant: The matrix is non-singular, allowing us to find the matrix inverse and hence solve for \(s\) and \(t\).

Considering this, the determinant acts as a gatekeeper, affirming the conditions necessary to utilize matrix inversion and guaranteeing that solutions exist and are unique.