Linear Algebra 1:5 Flashcards

Homogeneous Linear System

• Formulated as Ax = 0, where A is an m x n matrix and 0 is the zero vector in R(m).
• Always has at least one solution, known as the trivial solution.

Trivial and Nontrivial Solutions

• Trivial solution is the zero vector, satisfying Ax = 0.
• Nontrivial solution refers to any nonzero vector x that also satisfies Ax = 0.

Nontrivial Solutions Condition

• A nontrivial solution exists if and only if the equation has at least one free variable.

Zero Entries in Nontrivial Solutions

• Nontrivial solution x can contain zero entries but cannot be entirely composed of zeros.

Span of Zero Vector

• Span{0} only produces the zero vector as the solution.

Solutions with One Free Variable

• If Ax = 0 has a single free variable, the solution set represents a line through the origin.

Parametric Vector Equation

• Describes a plane spanned by vectors u and v as x = su + tv, where s and t are real numbers.

Parametric Vector Form

• When solutions are explicitly represented using vectors, they are considered in parametric vector form.

Nonhomogeneous System Solutions

• General solution for a nonhomogeneous linear system with multiple solutions is written in parametric vector form as one vector plus linear combinations satisfying the corresponding homogeneous system.

Vector Translation Concept

• Solution set of Ax - b can be visualized geometrically by considering vector addition as a translation, where vector v is translated by vector p to yield v + p.

Solution Set Characteristics

• The solution set of Ax = b is a line through point p, parallel to the solution set of Ax = 0.

Theorem 6 on Solution Sets

• If Ax = b is consistent for a specific b and p is a solution, the solution set is given by all vectors of the form w = p + v(h), where v(h) solves Ax = 0.
• Applicable only when Ax = b has at least one nonzero solution p.

Steps for Writing Solution Set in Parametric Vector Form

• Row-reduce the augmented matrix to its reduced echelon form.
• Express each basic variable in terms of free variables present in the equations.
• Represent a typical solution x as a vector dependent on any free variables.
• Decompose x into a linear combination of vectors using the free variables as parameters.

Test your knowledge of key concepts in Linear Algebra, focusing on homogeneous linear systems and solutions. Review important definitions and distinctions between trivial and nontrivial solutions through engaging flashcards.