Math Discussion: Linear System.
- What is the geometric sense of a linear system in two variables and a linear system in three variables?
A system with two variables has linear equations that determine a line on the x and y plane. Since the equations are linear they describe lines. Because a solution to a linear system must satisfy all the equations, the solution is the intersection of these lines. This could be either a line, a single point, or an empty set. For three variables each linear equation determines a plane in a 3D space, and the solution set is the intersection of these planes. This means the solution may be a plane, a line, a single point, or an empty set
- How do we use determinants for solving a system of linear equations?
Cramer’s rule allows us to find the solution by using determinants. The main determinant is the coefficient matrix. This is put as D. Then there is a determinant for each function. Like D sub x, D sub y, D sub z, etc. You must make a column. After you find the determinant of the coefficient matrix you must replace the first column of the coefficient matrix with constant numbers to find the determinant. Then you make a second column for y and so on.
- Describe in your own words the meaning of inverse matrix.
Matrix B is uniquely determined by A and is called an inverse of A and at this point is denoted. A square matrix is singular if the determinant is 0. A square matrix that is not invertible is called a singular or degenerate.
Sample responses
1. In regards to linear equations with two variables in the geometric sense, we can tell right away that this equation exists on two planes the X and Y place when the two meet at a single point we use that as a solution to the linear equation. In linear equations with three variables, we have three planes the X, Y, and Z, this becomes a bit more difficult to find a solution for as many points might meet in two planes but are not solutions to the equation unless they meet on all three planes. 2. We use determinants for solving a system of equations by making it easier to view what values need to be multiplied by one another and from there if the answers need to be subtracted or added to ultimately get the answer of the determinant. Especially ones that have three variables. It simplifies the system visually in order to mitigate any mistakes one may make if it was shown another way. 3. The inverse of a matrix is for example A-1 which in turn would give us the identity matrix in the end. We know it turns into an identity matrix when we reach a matrix that is 2×2 in size or in that same pattern 6×6, 5×5, etc. When getting the inverse matrix you must switch the rightmost values with each other and give the new top value on the right side a negative as well as the bottom left value. This begins the process of finding the inverse then you solve as required and you would ultimately end up with the identity matrix.