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Chapter 1: Introduction

Cross Products and Vectors

• The expression v cross w equal to the zero vector occurs when vectors v and w are parallel.

• If v cross w is not the zero vector, it is always equal to the negative of v cross b.

• For vectors b and w to ensure that b cross w results in the zero vector, they must be parallel where one is a scalar multiple of the other.

• When b and w are parallel, the area of the parallelogram formed by b and w is zero; hence, their cross product length is zero.

Data Science and Linear Algebra

• Data Science integrates statistics, computer science, and mechanics, relying heavily on linear algebra.

• Applications of linear algebra include Google's AI software, circuit analysis in electrical engineering, and optimization problems in various fields.

• It's foundational in solving large datasets, traffic flow analysis, and even in genetic studies and control systems in physics.

Introduction to Linear Equations

• Linear equation: Form where each term is a constant or a constant multiplied by the variable to the first power.

• Standard form: Variables on the left, constant on the right, ensuring variables are only to the first power, e.g., a₁x + a₂y = b.

Chapter 2: System Of Linear Equations

Identifying Linear Equations

• Examples of linear equations include:

  • First: 2x + y = 6 (standard form).

  • Second: x + y - z = 3 (not in standard form but can be rewritten).

  • Third: 2x + 3x - 4y = 0 (in standard form).

Nonlinear Equations

• Equations such as x² + y = 3 (quadratic), √y = x, and sin(x) + z = 1 are not linear due to polynomial and transcendental terms.

Systems of Linear Equations

• Linear system: Set of linear equations that must hold true simultaneously. For example, two equations create a 2x2 system.

• The circuit problems in electrical engineering lead to systems of linear equations where Kirchhoff's laws apply. An overdetermined system is one with more equations than variables.

Chapter 3: Simple Linear System

Solving Linear Systems

• Real-world applications often lead to complex systems requiring systematic methods similar to those used in school.

• Consistent systems: Have at least one solution; inconsistent systems: have no solutions. Examples from one-by-one systems show three cases:

  • One solution (e.g., x = 2/3).

  • No solutions (e.g., 0x = 2).

  • Infinitely many solutions (e.g., 0x = 0).

Chapter 4: The General Solution

Nature of Solutions in Linear Systems

• The general solution encompasses all values satisfying the equation:

  • Example: For 0x = 0, all real numbers are solutions.

• Case 1: One solution occurs when coefficients are non-zero.

• Case 2: Infinitely many solutions when both coefficients are zero.

• Case 3: No solution when the constant is non-zero while the coefficient is zero.

Chapter 5: Single Linear Equation

Characteristics of Linear Equations

• A one-by-two system (one equation, two variables) represents a line.

• There can never be a single solution in such systems, although cases of no solutions can arise from more complex systems.

• Graphically: The intersection points of lines represent solutions; two intersecting lines yield one solution, parallel lines yield none, while coincident lines yield infinitely many.

Chapter 6: General Solutions in 2D Systems

Solution Cases

• In two-variable systems, solutions can vary:

  • Single solution: Intersecting lines.

  • No solutions: Parallel lines.

  • Infinitely many solutions: Coincident lines.

• The general solution for a system of equations is all the ordered pairs (x, y) that satisfy both equations.

Chapter 7: Systems in Higher Dimensions

One-by-Three Systems

• Systems with one equation and three variables denote a plane.

• The vector parametric form for such a system highlights the relationship among variables, yielding solutions defined through parameters.

• The correctness of solutions hinges on representing variables aptly and acknowledging free variables which allow for infinite solutions.