Question

A matrix \(A\) and vector \(\vec{b}\) are given. Solve the equation \(A \vec{x}=\vec{b},\) write the solution in vector format, and sketch the solution as the appropriate line on the Cartesian plane. $$ A=\left[\begin{array}{cc} 2 & 4 \\ -1 & -2 \end{array}\right], \vec{b}=\left[\begin{array}{c} -6 \\ 3 \end{array}\right] $$

Short answer

Infinite solutions: \( \begin{bmatrix}-3-2t \ t \end{bmatrix} \), line through \((-3, 0)\) and \((-5, 1)\).

Step-by-step solution

Define the System of Equations

The matrix equation \( A\vec{x} = \vec{b} \) can be written as a system of equations by multiplying the matrix \( A \) with the vector \( \vec{x} \) and equating it to \( \vec{b} \). This results in: \[ 2x_1 + 4x_2 = -6 \] \[ -x_1 - 2x_2 = 3 \]

Solve the System of Equations

Let's solve the system: 1. Multiply the first equation by 1/2: - \( x_1 + 2x_2 = -3 \) 2. The second equation is the same: - \( -x_1 - 2x_2 = 3 \) 3. Add the two equations: - \( 0 = 0 \), suggesting the system is dependent and has an infinite number of solutions.

Express the General Solution

Since the system is dependent, we can express \( x_1 \) in terms of \( x_2 \). Utilizing the simplified first equation: \[ x_1 = -3 - 2x_2 \] Thus, the solution in vector form is: \[ \vec{x} = \begin{bmatrix}-3 - 2t \ t \end{bmatrix} \] where \( t \) is a real number.

Sketch the Solution on the Cartesian Plane

The solution set is a line determined by the parameter \( t \). We can sketch this line by selecting values for \( t \): 1. When \( t = 0 \), \( \vec{x} = \begin{bmatrix} -3 \ 0 \end{bmatrix} \). 2. When \( t = 1 \), \( \vec{x} = \begin{bmatrix} -5 \ 1 \end{bmatrix} \). 3. Draw a line through these points to represent the solution set on the plane.

Key concepts

Matrix Algebra

Matrix algebra is a type of mathematics that deals with matrices, which are rectangular arrays of numbers arranged in rows and columns. These can be used to represent and solve systems of linear equations. When working with matrices, especially in the context of equations, we use operations like addition, subtraction, multiplication, and scalar multiplication.

In our problem, we have a matrix \( A \) and a vector \( \vec{b} \). The equation \( A\vec{x} = \vec{b} \) involves matrix multiplication, where each element of the resulting vector is a sum of products. This concept is key to rewriting a system of equations in a compact form.

Understanding matrix operations is crucial, as they allow us to handle large systems of equations efficiently. This becomes even more important in fields like computer science and engineering, where such operations are routine.

• Matrices help organize complex systems into a manageable form.
• They facilitate the use of techniques like elimination and substitution systematically.
• Matrix algebra forms the backbone of many advanced mathematical concepts.

Dependent System

A dependent system of linear equations is a scenario where the equations in the system are not independent of each other. This means one equation can be derived from the others, indicating that they describe the same geometric object, like a line.

In our exercise, after simplifying the equations, we noticed that adding the two resulted in \( 0 = 0 \). This is a classic indication of dependency — there's no contradiction, but there's no new information either. Every equation essentially boils down to the same line, showing that the equations are dependent.

Here are some key properties of dependent systems:

• They typically have either no solution or infinitely many solutions.
• This happens when two or more equations in the set are directly related.
• Dependency can also stem from redundancy among equations.

Recognizing when a system is dependent helps in predicting the nature of its solutions and the kinds of mathematical manipulations that can simplify solving these problems.

Infinite Solutions

A system of equations has infinite solutions when there are limitless ways to satisfy the equations. Essentially, infinite solutions arise when the system is dependent.

In this exercise, after simplifying and manipulating the equations, we discovered that both represent the same line in space. Any point on that line is a solution to the system, thus resulting in infinite solutions.

When dealing with infinite solutions:

• Solutions can be expressed in terms of free variables, which are parameters like \( t \) in our example.
• The general solution gives a family of solutions that can be tested by plugging in values for the parameter.
• This kind of solution is often represented algebraically, adopting a vector form that incorporates the parameter.

Understanding infinite solutions helps in recognizing the geometric interpretation of systems, often showing up as lines or planes in graphical representations.

Vector Representation

Vector representation is a powerful way to express solutions to systems of linear equations, especially when infinite solutions are involved.

In the provided exercise, we expressed the solution in vector form as \[ \vec{x} = \begin{bmatrix} -3 - 2t \ t \end{bmatrix} \], where \( t \) is a real number. This form shows how each solution corresponds to a point on the line represented by these equations.

Vectors provide a clear geometric interpretation by representing points in space. Here are some benefits of using vector representation:

• It offers a concise form to express solutions, especially for complex systems.
• The parameter \( t \) showcases how the solution can vary, mapping an entire set of solutions.
• This method aligns closely with geometrical concepts, as vectors can directly represent lines or planes.

Recognizing and using vector representation is crucial for students learning linear algebra, as it bridges the gap between algebraic manipulation and geometric intuition.