Discussion – Linear Equations

Discussion – Linear Equations

Question 1

Two linear equations represent two lines in a plane, one line for each equation. The solution to the system of linear equations is the point at which the two lines intersect. If they cross at a single point, the system has a unique solution. If they never cross, the system has no solution. If the lines are just one, there are infinitely many solutions: every one of them lies on the line and satisfies both equations. The geometric perspective helps one see the kind of solution to the system: one point of intersection, no intersection, or two lines coinciding.

The representation of a linear system in three variables is three planes in three-dimensional space. The solution of the system is the point where they all intersect. If they intersect at a single point, there is exactly one solution to the system. If they intersect along a line, there are infinitely many. If no two of the planes are parallel or have a shared non-trivial intersection, there is no solution to the system. The relationship of the planes and the number and nature of their intersections makes it easier to get a handle on in three dimensions, which is why geometers prefer to work it out that way.

Question 2

Determinants are used to solve systems of linear equations by methods such as Cramer’s Rule. In a system where the coefficient matrix has a non-zero determinant, there is a unique solution. One can find the values of the variables by computing the determinant of several matrices, where one column of the matrix is the constant from the corresponding equations. On a small system, it gives one a direct and very ‘algebraic’ approach to solving problems related to the properties of determinants.

Question 3

An inverse matrix is essentially the reciprocal of a matrix, denoted as A⁻¹ for a matrix A. When a matrix is multiplied by its inverse, the result is the identity matrix, analogous to multiplying a number by its reciprocal to get one. In the context of solving linear equations, if the coefficient matrix A has an inverse, the system Ax=b can be solved by multiplying both sides by A⁻¹, yielding x=A⁻¹b (Kazunga & Bansilal, 2020). This provides a direct method for finding the solution vector x, making the concept of an inverse matrix fundamental in linear algebra.

References

Kazunga, C., & Bansilal, S. (2020). An APOS analysis of solving systems of equations using the inverse matrix method. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-020-09935-6

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Linear Equations

Please write one or two paragraphs on each topic. Express your vision of the topic in your own words.

1. What is the geometric sense of a linear system in two variables and a linear system in three variables?
2. How do we use determinants for solving a system of linear equations?
3. Describe in your own words the meaning of an inverse matrix.