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Linear Algebra For Undergraduate Students

Yilan(Peter) Tan

Section 1.2 Linear Systems and Equivalent Systems

Definition 1.2.1. Linear Equation.

A linear equation in the variables \(x_1, x_2, \dots, x_n\) is an equation of the form
\begin{equation*} a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b \end{equation*}
where the coefficients \(a_1,\dots,a_n\) and the constant \(b\in \mathbb{R}\text{.}\) In a linear equation each variable appears only to the first power and no products of variables occur. A solution is an ordered n-tuple \((x_1,\dots,x_n)\) that satisfies the equation.
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Exercise: Describe the geometric representation of solutions to linear equations in the plane and in space.
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  1. In the plane: Consider the linear equation in two-dimensional space \(\mathbb{R}^2\)
    \begin{equation*} 2x_1 + 3x_2 = 6 \end{equation*}
    The solution set of this equation geometrically represents a line. We can find several particular solutions:
    • When \(x_1 = 0\text{,}\) we have \(x_2 = 2\text{,}\) giving the point \((0, 2)\)
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    • When \(x_1 = 3\text{,}\) we have \(x_2 = 0\text{,}\) giving the point \((3, 0)\)
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    • When \(x_1 = -3\text{,}\) we have \(x_2 = 4\text{,}\) giving the point \((-3, 4)\)
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    All these points lie on the same line, which is the solution set of the equation.
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  2. In space: Consider the linear equation in three-dimensional space \(\mathbb{R}^3\)
    \begin{equation*} x_1 + 2x_2 + 3x_3 = 12 \end{equation*}
    The solution set of this equation geometrically represents a plane. We can find several particular solutions:
    • When \(x_1 = 12, x_2 = 0, x_3 = 0\text{,}\) we get the point \((12, 0, 0)\)
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    • When \(x_1 = 0, x_2 = 6, x_3 = 0\text{,}\) we get the point \((0, 6, 0)\)
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    • When \(x_1 = 0, x_2 = 0, x_3 = 4\text{,}\) we get the point \((0, 0, 4)\)
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    • When \(x_1 = 6, x_2 = 3, x_3 = 0\text{,}\) we get the point \((6, 3, 0)\)
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    All these points lie on the same plane, which is the solution set of the equation.
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Definition 1.2.2.

A linear system (or system of linear equations) is a collection of linear equations involving the same set of variables.
\begin{equation*} \begin{aligned} a_{11}x_1 + a_{12}x_2 +&\ldots+ a_{1n}x_n = b_1\\ a_{21}x_1 + a_{22}x_2 +&\ldots+ a_{2n}x_n = b_2\\ &\quad \vdots & \\ a_{m1}x_1 + a_{m2}x_2 +&\ldots+ a_{mn}x_n = b_m\\ \end{aligned} \end{equation*}
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A solution to the linear system is an ordered n-tuple \((x_1, x_2, \dots, x_n)\) that satisfies all equations in the system simultaneously. A solution set of the linear system is the collection of all possible solutions. A Linear system is called consistent if it has at least one solution; otherwise, it is called inconsistent.
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For example, consider the following system of three equations in three variables:
\begin{equation*} \begin{aligned} 2x + 3y - z &= 7\\ x - y + 2z &= 4\\ 3x + y + z &= 8 \end{aligned} \end{equation*}
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Study Goal of the chapter.

Our goal of this chapter is to find the solution set of a linear system.
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Two linear systems are called equivalent if they have exactly the same set of solutions. The key insight in solving linear systems is that we can transform a given system into an equivalent system that is easier to solve, without changing the solution set.
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The transformation from one system to an equivalent system is accomplished through elementary equation operations:
  1. Multiply an equation by a non-zero constant.
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  2. Add a multiple of one equation to another equation.
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  3. Interchange two equations.
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These operations preserve the solution set, meaning that if we apply any combination of these operations to a linear system, the resulting system will be equivalent to the original system, that is, they have the same solution set.
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Example 1.2.3. Elementary Equation Operations and Equivalent Systems.

Let’s demonstrate each elementary equation operation using the system:
\begin{equation*} \begin{aligned} x + 2y &= 5\\ 2x + 3y &= 7 \end{aligned} \end{equation*}
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Operation 1: Multiply an equation by a non-zero constant
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We can multiply the first equation by 2 to get an equivalent system:
\begin{equation*} \begin{aligned} 2x + 4y &= 10\\ 2x + 3y &= 7 \end{aligned} \end{equation*}
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Notice that we can reverse this operation by dividing the first equation by 2 to return to the original system. The transformation works both ways.
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Operation 2: Add a multiple of one equation to another
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Starting from our original system, we subtract twice the first equation from the second equation:
\begin{equation*} \begin{aligned} x + 2y &= 5\\ -y &= -3 \end{aligned} \end{equation*}
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We can reverse this by adding twice the first equation to the second equation. Again, the transformation is reversible.
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Operation 3: Interchange two equations
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We can swap the order of the equations:
\begin{equation*} \begin{aligned} 2x + 3y &= 7\\ x + 2y &= 5 \end{aligned} \end{equation*}
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Obviously, we can interchange them back to the original order. This operation is clearly reversible.
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The key insight is that every elementary equation operation can be undone. This guarantees that when we transform one system into another using these operations, we can always transform back, ensuring the systems are truly equivalent with the same solution set.
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