Chapter 1 System of linear equations
Solving systems of linear equations is a fundamental technique in matrix theory and forms the backbone of many mathematical and scientific applications. By representing a system of equations as a matrix equation, we can utilize powerful matrix operations to analyze and solve complex problems efficiently. This approach not only streamlines computations but also provides deeper insights into the structure and properties of linear systems. Understanding how to manipulate matrices to find solutions enables us to tackle a wide range of challenges in engineering, physics, computer science, and economics, making it an essential skill for anyone studying linear algebra and matrix theory.
Example: Let
\begin{equation*}
\begin{aligned}
\mathbf{a}_1 &= \begin{bmatrix}0\\1\\3\\0\end{bmatrix}, &
\mathbf{a}_2 &= \begin{bmatrix}1\\0\\0\\-1\end{bmatrix}, &
\mathbf{a}_3 &= \begin{bmatrix}-3\\3\\5\\3\end{bmatrix}, &
\mathbf{a}_4 &= \begin{bmatrix}-4\\2\\6\\10\end{bmatrix}, &
\mathbf{b} &= \begin{bmatrix}15\\-3\\-17\\-39\end{bmatrix}.
\end{aligned}
\end{equation*}
Solve the vector equation
\begin{equation*}
x_1\mathbf{a}_1 + x_2\mathbf{a}_2 + x_3\mathbf{a}_3 + x_4\mathbf{a}_4 = \mathbf{b}.
\end{equation*}
That is, find scalars \(x_1,x_2,x_3,x_4\) such that the vector equation holds.
