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Chapter 1: Introduction

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 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.

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.

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).

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

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.

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).

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.

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.

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.

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.