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A system of equations is a collection of two or more equations with a common set of variables. These equations require a solution that satisfies every equation in the system. In the context of the exercise, the system of equations is derived from the expression \(Ax = b\). Here, each row of matrix \(A\) forms an equation, contributing to a complete system of equations. The main task is to find values for the vector \(x\) that satisfies all these equations simultaneously.

When expressing the column matrix \(b\) as a linear combination of the columns of \(A\), a system of equations is essentially established. Each component of \(b\) corresponds to a combination derived from the values in matrix \(A\), multiplied by components of \(x\). Solving this system involves:

• Identifying each equation formed from the rows.
• Solving the equations to find the values of \(x_1, x_2,\) and \(x_3\).

These solutions are crucial in understanding how linear combinations work in the context of matrices.

A column matrix is simply a matrix consisting of a single column of numbers. It represents a vector in matrix terminology and can be an integral part of linear algebra calculations.

In the exercise, the column matrix \(b\) is represented as \(\begin{bmatrix}3\1\0\end{bmatrix}\). This means it has entries, or elements, arranged vertically with three rows. Each row corresponds to a separate equation when breaking down the system of equations from the previous step.

Column matrices are essential in matrices operations involving systems of linear equations and in linear combinations, as each column matrix can represent a singular variable or outcome in the context of vector spaces.

Matrix multiplication is a fundamental operation in linear algebra where two matrices are multiplied to produce another matrix. This involves the dot product of rows of the first matrix and columns of the second matrix.

In the context of the problem, calculating \(Ax\) involves multiplying matrix \(A\) with vector \(x\). This multiplication results in a new column matrix that we set equal to \(b\). The components of the resulting matrix are made by:

• Multiplying corresponding entries across rows of \(A\) and columns of \(x\).
• Summing those products to form each element of the column matrix.

Matrix multiplication is crucial because it allows for the formulation and solving of systems of equations, turning a seemingly complex problem into a set of simpler calculations. This method is widely used to solve linear algebra problems and model real-world systems in engineering, physics, computer science, and economics.