Linear Algebra Learning Links: Consistency, Matrices, Parallel, Intersecting, Identical Lines and Planes

Consistency, independence and dependency are more easily understood from a geometrically standpoint than a algebraical standpoint.

Two linear equations that have different slopes will intersect at a point. The two equations are said to be linearly independent and consistent (they have exactly one solution).

A 2D Representation of an Object in 3D Space

Duck Chess Piece, Ancient Rome

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Two linear equations may have the same slope but different y-intercepts, that is they are parallel. These two linear equations are said to be inconsistent, that is they have no solution(they do not intersect).

Two linear equations may have the same multiple of  the slope and the y-intercept (e.g. y = x + 3 and 3y = 3x + 9). That is they are the same line. These two linear equations are said to be linearly dependent.

These definitions can be extended to planes in 3D space.

When linear systems are placed in a matrix, the consistency, independence and dependence of the equations can be determined from the value of the determinant, the linear dependence between rows and the consistency between columns.  If a row is a multiple of another row, than the equations represent identical  geometrys at the same location(linearly dependent lines, planes and objects). If two rows are identical except for the constant(intercept term), than the geometrical(graphical) representation of the equations the rows represent are parallel objects (parallel lines, planes and forms). These objects do not intersect and hence have no solution and are inconsistent.

Library Learning Links
http://www.algebra.com/algebra/homework/coordinate/Types-of-systems-inconsistent-dependent-independent.lesson

http://math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/system/system.html