13.3 Systems of linear equations

Systems of linear equations are the backbone of linear algebra, connecting various concepts like matrices and determinants. They're used to model real-world problems and solve complex equations simultaneously.

Understanding these systems is crucial for tackling more advanced topics in linear algebra. We'll look at different ways to represent and solve them, from matrix forms to elimination methods and special system types.

Matrix Representations

Matrix Forms

Top images from around the web for Matrix Forms

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

1 of 3

Top images from around the web for Matrix Forms

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

• 

  3.5b. Examples – Augmented Matrices | Finite Math View original

  Is this image relevant?

1 of 3

• Coefficient matrix represents the coefficients of the variables in a system of linear equations
  • Formed by extracting the coefficients from each equation and arranging them in a matrix
  • Does not include the constants on the right-hand side of the equations
• Augmented matrix combines the coefficient matrix and the constants from the right-hand side of the equations
  • Formed by appending an additional column to the coefficient matrix, containing the constants
  • Provides a compact representation of the entire system of linear equations

Echelon Forms

• Row echelon form is a matrix form where the leading entry (first nonzero number from the left) of a row is always strictly to the right of the leading entry of the row above it
  • All entries below a leading entry are zeros
  • Obtained through a sequence of elementary row operations (row switching, row multiplication, row addition)
• Reduced row echelon form is a row echelon form with additional conditions
  • Leading entry in each row is 1
  • Each column containing a leading 1 has zeros in all its other entries
  • Obtained by performing additional row operations on a matrix in row echelon form

Solution Methods

Gaussian Elimination

• Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix into row echelon form
  • Involves performing a sequence of elementary row operations to eliminate variables and obtain a triangular matrix
  • The solution is then found by back-substitution, starting from the bottom equation and working upwards
• The steps in Gaussian elimination are:
  1. Write the system of equations as an augmented matrix
  2. Use elementary row operations to transform the matrix into row echelon form
  3. Use back-substitution to find the values of the variables

Cramer's Rule

• Cramer's rule is a formula for solving systems of linear equations using determinants
  • Expresses the solution in terms of the determinants of the coefficient matrix and matrices obtained by replacing one column of the coefficient matrix with the constants
• The formula for Cramer's rule is:
  • $x_i = \frac{det(A_i)}{det(A)}$, where $A$ is the coefficient matrix, $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the constants, and $det$ denotes the determinant
• Cramer's rule is practical for small systems but becomes computationally inefficient for large systems due to the calculation of determinants

System Types

Homogeneous Systems

• A homogeneous system of linear equations is a system where all the constant terms (right-hand side) are zero
  • Has the form $a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = 0$, $a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = 0$, ..., $a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = 0$
  • Always has at least one solution, the trivial solution, where all variables are zero $(x_1, x_2, ..., x_n) = (0, 0, ..., 0)$
• A homogeneous system may have non-trivial solutions depending on the rank of the coefficient matrix
  • If the rank of the coefficient matrix is equal to the number of variables, the system has only the trivial solution
  • If the rank is less than the number of variables, the system has infinitely many solutions

Nonhomogeneous Systems

• A nonhomogeneous system of linear equations is a system where at least one of the constant terms (right-hand side) is nonzero
  • Has the form $a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1$, $a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2$, ..., $a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m$, where at least one $b_i$ is nonzero
• The existence and uniqueness of solutions for a nonhomogeneous system depend on the relationship between the rank of the coefficient matrix and the rank of the augmented matrix
  • If the ranks are equal and equal to the number of variables, the system has a unique solution
  • If the ranks are equal but less than the number of variables, the system has infinitely many solutions
  • If the rank of the coefficient matrix is less than the rank of the augmented matrix, the system has no solution (inconsistent system)