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.