Introduction to Linear Systems

Introduction to Linear Systems

• Discussion on solving systems of linear equations.

Definition of a Linear System

• A system of m linear equations with n unknowns $x_1, x_2, ..., x_n$ is a set of equations in the form:
  • $a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n = b_1$
  • $a_{21}x_1 + a_{22}x_2 + ... + a_{2n}x_n = b_2$
  • $a_{m1}x_1 + a_{m2}x_2 + ... + a_{mn}x_n = b_m$
• $a_{ij}$ and $b_i$ are given scalars.

Solution of a Linear System

• A solution is a list of n scalars $s_1, s_2, ..., s_n$ that satisfies each equation when substituted for $x_1 = s_1, x_2 = s_2, ..., x_n = s_n$.

Geometric Interpretation: Two Equations and Two Unknowns

• Given the system:
  • $a_{11}x_1 + a_{12}x_2 = b_1$
  • $a_{21}x_1 + a_{22}x_2 = b_2$
• Each equation represents a line in the plane.
• The solution is the intersection of these two lines.

Example 2.1

• Solve the system:
  • $x_1 - x_2 = 1$
  • $x_1 + x_2 = 3$
• By adding the equations, $2x_1 = 4$, so $x_1 = 2$.
• Substituting into the first equation, $2 - x_2 = 1$, so $x_2 = 1$.
• The solution is $(2, 1)$.

Geometric Interpretation: Three Equations and Three Unknowns

• Given the system:
  • $a_{11}x_1 + a_{12}x_2 + a_{13}x_3 = b_1$
  • $a_{21}x_1 + a_{22}x_2 + a_{23}x_3 = b_2$
  • $a_{31}x_1 + a_{32}x_2 + a_{33}x_3 = b_3$
• Each equation represents a plane in space.
• The solution is the intersection of these three planes.

Example 2.2

• Solve the system:
  • $x_1 - x_2 + x_3 = 2$
  • $2x_1 - x_2 + x_3 = 3$
  • $-x_1 + x_2 + x_3 = 4$
• Adding the first and third equations, $2x_3 = 6$, so $x_3 = 3$.
• Substituting into the first equation, $x_1 - x_2 + 3 = 2$, so $x_1 - x_2 = -1$.
• Substituting into the second equation, $2x_1 - x_2 + 3 = 3$, so $2x_1 - x_2 = 0$.
• Subtracting the first equation from the second, $x_1 = 1$.
• Substituting into the first equation, $1 - x_2 = -1$, so $x_2 = 2$.
• The solution is $(1, 2, 3)$.

Solving Linear Systems: Elementary Operations

• Elementary operations on the equations of a linear system include:
  • Interchanging two equations.
  • Multiplying an equation by a non-zero scalar.
  • Adding a multiple of one equation to another.

Echelon Form

• Transforming a system into an equivalent system that is easier to solve.

Definition 2.3: Echelon Form

• A linear system is in echelon form if:
  • The first non-zero coefficient (from the left) of each equation (called the pivot) is to the right of the pivot of the previous equation.
  • Equations with all coefficients equal to zero are grouped at the bottom of the system.

Solving by Substitution

• Once the system is in echelon form, solve by substitution starting from the last equation and working upwards.

Example 2.3

• Solve the system:
  • $x_1 - x_2 + x_3 = 2$
  • $x_2 - x_3 = 1$
  • $x_3 = 3$
• Substituting $x_3 = 3$ into the second equation, $x_2 - 3 = 1$, so $x_2 = 4$.
• Substituting $x_2 = 4$ and $x_3 = 3$ into the first equation, $x_1 - 4 + 3 = 2$, so $x_1 = 3$.
• The solution is $(3, 4, 3)$.

Gauss-Jordan Elimination

Definition 2.4: Reduced Echelon Form

• A linear system is in reduced echelon form if:
  • The system is in echelon form.
  • The first non-zero coefficient of each equation is 1.
  • This 1 is the only non-zero element in its column.

Example 2.4

• Solve the system:
  • $x_1 - x_2 + x_3 = 2$
  • $2x_1 - x_2 + x_3 = 3$
  • $-x_1 + x_2 + x_3 = 4$
• Subtracting 2 times the first row from the second, and adding the first row to the third:
  • $x_1 - x_2 + x_3 = 2$
  • $x_2 - x_3 = -1$
  • $2x_3 = 6$
• Dividing the third row by 2:
  • $x_1 - x_2 + x_3 = 2$
  • $x_2 - x_3 = -1$
  • $x_3 = 3$
• Adding the second row to the first, and adding the third row to the second:
  • $x_1 + x_3 = 1$
  • $x_2 = 2$
  • $x_3 = 3$
• Subtracting the third row from the first:
  • $x_1 = -2$
  • $x_2 = 2$
  • $x_3 = 3$
• The solution is $(-2, 2, 3)$.

Matrices

Definition 2.5

• A matrix is a rectangular array of numbers.
• The numbers are called the elements of the matrix.
• A matrix with m rows and n columns is called an $m \times n$ matrix.

Example 2.5

• $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \end{bmatrix}$ is a $2 \times 3$ matrix.

Operations on Matrices

Addition

• Add two matrices of the same size by adding corresponding elements.

Scalar Multiplication

• Multiply a matrix by a scalar by multiplying each element by the scalar.

Matrix Multiplication

• Multiply two matrices A and B if the number of columns of A equals the number of rows of B.
• If A is an $m \times n$ matrix and B is an $n \times p$ matrix, then the product AB is an $m \times p$ matrix.
• Elements are given by $(AB){ij} = \sum{k=1}^{n} A_{ik}B_{kj}$.

Example 2.6

• Given $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}$. Compute AB.
• $AB = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \ 43 & 50 \end{bmatrix}$.