Question

In each case find a vector equation that is equivalent to the given system of equations. $$ \begin{array}{l} \text { a. } \quad x_{1}-x_{2}+3 x_{3}=5 \\ -3 x_{1}+x_{2}+x_{3}=-6 \\ 5 x_{1}-8 x_{2}=9 \\ \text { b. } \quad x_{1}-2 x_{2}-x_{3}+x_{4}=5 \\ -x_{1}+x_{3}-2 x_{4}=-3 \\ 2 x_{1}-2 x_{2}+7 x_{3}=8 \\ 3 x_{1}-4 x_{2}+9 x_{3}-2 x_{4}=12 \end{array} $$

Short answer

The vector equations are: a: \(A\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 5 \\ -6 \\ 9 \end{bmatrix}\), b: \(A\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 5 \\ -3 \\ 8 \\ 12 \end{bmatrix}\).

Step-by-step solution

Express Each Equation in Vector Form - Part a

Let's start by writing the given system of equations in a vector form:The system is:1. \(x_1 - x_2 + 3x_3 = 5\)2. \(-3x_1 + x_2 + x_3 = -6\)3. \(5x_1 - 8x_2 = 9\)We can express it as a vector equation of the form \(Ax = b\), where \(A\) is the coefficient matrix, \(x\) is the vector of variables, and \(b\) is the constants vector:\[\begin{bmatrix}1 & -1 & 3 \-3 & 1 & 1 \5 & -8 & 0\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix} =\begin{bmatrix}5 \-6 \9\end{bmatrix}\]

Express the Solution as Vector Equation - Part a

Now, rewrite the matrix-vector multiplication from Step 1 as a vector equation:\[ \begin{bmatrix} 1 \ x_1 \end{bmatrix} + \begin{bmatrix} -1 \ x_2 \end{bmatrix} + \begin{bmatrix} 3 \ x_3 \end{bmatrix} = \begin{bmatrix} 5 \ -6 \ 9 \end{bmatrix} \]

Express Each Equation in Vector Form - Part b

The given system is:1. \(x_1 - 2x_2 - x_3 + x_4 = 5\)2. \(-x_1 + x_3 - 2x_4 = -3\)3. \(2x_1 - 2x_2 + 7x_3 = 8\)4. \(3x_1 - 4x_2 + 9x_3 - 2x_4 = 12\)We express this as a vector equation \(Ax = b\):\[\begin{bmatrix}1 & -2 & -1 & 1 \-1 & 0 & 1 & -2 \2 & -2 & 7 & 0 \3 & -4 & 9 & -2\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3 \x_4\end{bmatrix} =\begin{bmatrix}5 \-3 \8 \12\end{bmatrix}\]

Express the Solution as Vector Equation - Part b

Rewrite the matrix-vector multiplication from Step 3 in component form:\[ \begin{bmatrix}x_1 \x_2 \x_3 \x_4\end{bmatrix} = \begin{bmatrix}1 \-2 \-1 \1 \-1 \0 \1 \-2 \2 \-2 \7 \0 \3 \-4 \9 \-2\end{bmatrix} + \\begin{bmatrix} 5 \-3 \8 \12 \\end{bmatrix}\]

Key concepts

Linear Algebra

Linear algebra is a fascinating area of mathematics that primarily deals with vectors, vector spaces, and linear equations. It's used in various fields such as physics, computer science, and economics to model and solve real-world problems. One of the key aspects of linear algebra is understanding how vectors can express systems of linear equations. Vectors are essentially lists of numbers, which can represent points in space or directions.
For example, in a 3-dimensional space, any point can be represented by a vector \( [x, y, z] \), where each component corresponds to a coordinate axis.
By using vectors and the operations defined on them, we can perform vector addition, scalar multiplication, and dot products, which are fundamental concepts in solving systems of equations. Here, each equation of the system can be expressed using these operations, allowing a geometrical view of the solutions. These are often visualized as planes (or lines in two dimensions) with solutions corresponding to their intersections. Linear algebra helps to simplify these complex operations by organizing these simple vectors into more complex structures like matrices, which are a central concept of this mathematical field.

Matrix Representation

In the context of representing systems of linear equations, matrices are incredibly powerful tools. A matrix is essentially a rectangular array of numbers arranged in rows and columns. When solving systems of linear equations, each equation is a linear combination of variables that can be represented as rows in a matrix.
To represent the system of equations, we arrange the coefficients of the variables from each equation into a matrix. This is known as the coefficient matrix. For example, considering the exercise from above, the matrix representation for the system is given by matrices \( [A] \) and vectors \( [b] \), forming the equation \(Ax = b\).

• Each row in the matrix \(A\) corresponds to the coefficients of one equation.
• The vector \([x]\) contains the variables we're solving for.
• The vector \(b\) contains the constants from the equations.

Understanding matrix representation allows us to use various matrix operations like addition, subtraction, multiplication, and especially techniques like Gaussian elimination or other algorithmic approaches to find solutions. These are crucial in simplifying and solving complex systems of equations effectively.

System of Equations

A system of equations is a set of two or more equations with a shared set of unknowns. Solving systems of equations involves finding the values of these unknowns that satisfy all the given equations simultaneously.
System of equations can be expressed in different forms:

• Standard form - equations are given in their explicit algebraic form, like \(x_1 - x_2 + 3x_3 = 5\).
• Vector equation form - where equations are rewritten as vector expressions, depicting the relationship between vectors.
• Matrix-vector form as \(Ax = b\) - offering a compact representation of the entire system.

Solving a system often involves either graphical methods (though more challenging for systems in higher dimensions) or by algebraic methods like substitution, elimination, or utilizing matrix techniques like the inverse matrix method, if applicable. The solution could be a single point, a line of points, or no solution at all, depending on the nature of the system (consistent or inconsistent) and the number of equations and variables involved.
Understanding how these systems are represented and solved is fundamental in effectively addressing a multitude of real-world problems across various disciplines.