Elementary Linear Algebra 8th Edition - Chapter 1 Notes

• Linear Equations & Matrices: Organizing info in rows & columns as matrices.
• Solving linear equation systems using matrix operations; vital for computer solutions.
• Matrices have rich theory & broad applications; this chapter introduces their study.

Introduction to Systems of Linear Equations

• Linear Equations:

  • Line in xy-plane: $a₁x + a₂y = b$ ((a1, a2) not both zero).
  • General form in n variables: $a₁x₁ + a₂x₂ + … + a<em>nx</em>n = b$
  • Variables are unknowns.
  • Examples:
    • Linear: $x + 3y = 7, y = x + 3 + 1, x₁ - 2x₂ + 3x₃ + x₄ = 7$
    • Not linear: $x + 3√y = 5, 3x + 2y - z + xz = 4, y = sin x$
  • Solution: Sequence $s₁, s₂,…s_n$ satisfying the equation.
  • Solution set: All solutions (general solution).
    • Example 1: Assign x=t, get $y = 2t - \frac{1}{2}$. Assign y=t, get $x = \frac{1}{2} + \frac{1}{2}t$. Both are valid.
    • Example 2: Assign $x₂=s$ and $x₃=t$, get $x₁ = 5 + 4s - 7t$

• Linear Systems:

  • Finite set of linear equations.
  • Solution: Sequence satisfying all equations.
  • Inconsistent: No solutions.
  • Consistent: At least one solution.
  • Possibilities: No solution, one solution, or infinitely many solutions.
  • General system: Expressed with $a<em>{ij}$ and $b</em>i$
    • $a{11}x{1} + a{12}x{2} + … + a{1n}x{n} = b_1$
    • $a{21}x{1} + a{22}x{2} + … + a{2n}x{n} = b_2 - …$
    • $a{m1}x{1} + a{m2}x{2} + … + a{mn}x{n} = b_m$
  • Double subscripting: $a<em>{ij}$ indicates the coefficient of variable $x</em>j$ in equation $i$

Augmented Matrices

• Systems abbreviated as augmented matrices:

  • Coefficients and constants in rectangular array.
    $\begin{bmatrix} a<em>{11} & a</em>{12} & … & a<em>{1n} & b</em>1 \ a<em>{21} & a</em>{22} & … & a<em>{2n} & b</em>2 \ … & … & … & … & … \ a<em>{m1} & a</em>{m2} & … & a<em>{mn} & b</em>m \end{bmatrix}$
  • Unknowns in same order, constants on right.

• Solving via new systems with same solution set but which are easier to solve:

  • Multiply an equation by a nonzero constant; Interchange two equations; add a multiple of one equation to another; These operations correspond to elementary row operations on the augmented matrix

Gaussian Elimination

• Solving via Elementary Row Operations - systematic elimination:

  • Elementary row operations on augmented matrix provide efficient calculation to solve systems of linear equations

Echelon Forms

• Reduced Row-Echelon Form Properties:
  • Leading 1 in each row (if not all zeros).
  • Zero rows grouped at bottom.
  • Successive rows' leading 1s farther right.
  • Columns with leading 1s have zeros elsewhere.
• Row-Echelon Form:
  • Satisfies first three properties.
• Reduced form necessitates row-echelon form.
• Row-echelon form has zeros below leading 1s; reduced form has zeros above and below.
• Solve system easily from reduced row-echelon form.

Gaussian Elimination Example

• After reduction to reduced row-echelon form, solve by inspection.
  • If free variables exist, express leading variables in terms of free variables.
  • If a row of the form $[0 0 … 0 | 1]$ exists, the system has no solution.

Gauss-Jordan Elimination

• Gaussian elimination produces row-echelon form.
• Gauss-Jordan elimination continues to reduced row-echelon form.
• Gauss-Jordan avoids back-substitution; suitable for hand calculation but requires more operations on a computer.

Karl Friedrich Gauss

• (1777-1855) was a German mathematician and scientist. Sometimes called the "prince of mathematicians," Gauss ranks with Isaac Newton and Archimedes as one of the three greatest mathematicians who ever lived.

Wilhelm Jordan

• (1842-1899) was a German engineer who specialized in geodesy. His contribution to solving linear systems appeared in his popular book, Handbuch der Vermessungskunde (Handbook of Geodesy), in 1888.

Back-Substitution

• Using Gaussian elimination to row-echelon form, then back-substituting to solve.
• Solve equations for leading variables.
• Substitute from bottom up.
• Assign arbitrary values to free variables.

Homogeneous Linear Systems

• Constant terms are all zero.
• Always consistent (trivial solution).
• Solutions: Only the trivial solution or infinitely many solutions.
• More unknowns than equations: Nontrivial solutions exist.

Theorem 1.2.1

• A homogeneous system of linear equations with more unknowns than equations has infinitely many solutions.

Computer Solution of Linear Systems

• Based on Gaussian/Gauss-Jordan elimination.
• Considerations:
  • Reducing roundoff errors;
  • Minimizing memory use;
  • Maximizing speed.

Remarks

• Gauss-Jordan involves less writing, but Gaussian elimination requires fewer operations for large systems. Thus, Gaussian elimination is used on a computer.