A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix
In linear algebra, a linear system refers to a set of equations involving variables, which can be solved by finding their values
In linear algebra, a linear system refers to a set of equations involving variables, which can be solved by finding their values. The coefficient matrix is formed by the coefficients of the variables in the system of equations.
In a linear system, a pivot column is a column in the coefficient matrix that contains the leading entry (the first non-zero entry) of a row in row echelon form or reduced row echelon form. Pivot columns play a crucial role in solving linear systems as they determine the basic variables.
A basic variable is a variable that corresponds to a pivot column in the coefficient matrix. Essentially, it is a variable that directly contributes to determining the values of the other variables in the system. The value of a basic variable is directly determined by the value of the corresponding pivot column.
To illustrate this concept, let’s consider a linear system with three equations and three variables:
1) 2x + 3y – z = 5
2) -x + 6y + 2z = -1
3) 4x + 2y + 3z = 10
To solve this system, we form the coefficient matrix:
[ 2 3 -1 ]
[ -1 6 2 ]
[ 4 2 3 ]
Next, we perform row operations to create row echelon form or reduced row echelon form. Let’s assume the resulting matrix is:
[ 1 0 2 ]
[ 0 1 -1 ]
[ 0 0 0 ]
In this row echelon form, we observe that the leading entries (the pivots) are in the first and second columns. Thus, the first and second variables, x and y, are the basic variables. The third variable, z, which corresponds to the pivot column, is called a free variable.
The values of the basic variables can be found directly from the row echelon form. In this example, the value of x would be 2, and the value of y would be -1. The value of the free variable z can be assigned any real number, allowing for an infinite number of solutions.
So, in summary, a basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix. It directly contributes to determining the values of the other variables, while the value of a free variable can be assigned arbitrarily.
More Answers:
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Solving Systems of Linear Equations using the Row Reduction Algorithm: A Step-by-Step Guide