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Linear Algebra For Undergraduate Students

Yilan(Peter) Tan

Section 1.1 Geometric Interpretation: What Are We Really Solving?

In this course, we’ll solve systems of linear equations algebraically using matrices. But what does a system of equations really represent geometrically? Understanding the geometric meaning helps build intuition and reveals why certain systems have unique solutions, no solutions, or infinitely many solutions.
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Subsection 1.1.1 Three Equations in Three Variables: Planes in Space

For a \(3 \times 3\) system:
\begin{equation*} \begin{aligned} a_1x + b_1y + c_1z \amp= d_1\\ a_2x + b_2y + c_2z \amp= d_2\\ a_3x + b_3y + c_3z \amp= d_3 \end{aligned} \end{equation*}
Each equation represents a plane in 3D space. Solving means finding \((x, y, z)\) where all three planes meet.
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Shows different ways three planes can intersect: at a point, along a line, as a plane, or with no common intersection.
Figure 1.1.1. Possible Intersections of Three Planes
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In 3D, the geometry is richer:
  • Unique solution: Three planes meet at a single point (like the corner of a room).
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  • Infinitely many solutions (line): Three planes intersect along a common line.
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  • Infinitely many solutions (plane): All three equations describe the same plane!
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  • No solution: Planes don’t all meet at any common point. This happens when:
    • Some planes are parallel
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    • Planes form a "triangular prism" configuration
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Insight 1.1.2. Geometric Insight into Solutions.

The number of solutions to a linear system has geometric meaning:
  • Unique solution: Geometric objects intersect at exactly one point (general position).
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  • No solution: Objects are "parallel" or configured so they never all intersect.
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  • Infinitely many: Objects overlap more than expected (some equations redundant).
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Later, we’ll use matrix methods (row reduction) to determine which case we’re in, without needing to visualize high-dimensional geometry!
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Now that we understand the geometric picture, let’s develop algebraic tools to solve these systems systematically and efficiently, regardless of dimension!
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