Linear Algebra 1:5 Flashcards

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Questions and Answers

What is a homogeneous linear system?

A system of equations that has more variables than equations

A system of linear equations with only trivial solutions

A system of linear equations with no solutions

A system of linear equations that can be written in the form Ax=0 (correct)

A homogeneous linear system always has at least one solution.

True

What is the trivial solution?

The zero solution.

What is a nontrivial solution?

A nonzero vector x that satisfies Ax = 0. Signup and view all the answers

The homogeneous equation Ax=0 has a nontrivial solution if it has at least one free variable.

True Signup and view all the answers

A nontrivial solution x can only consist of nonzero entries.

False Signup and view all the answers

What is the ONLY solution in Span{0}?

The zero vector. Signup and view all the answers

If the equation Ax=0 has only one free variable, then the solution set is a line through the origin.

True Signup and view all the answers

What is a parametric vector equation?

An explicit description of the plane from the set spanned by u and v, written as x = su + tv. Signup and view all the answers

Whenever a solution is described explicitly with vectors, we say that the solution is in parametric vector form.

True Signup and view all the answers

What characterizes a nonhomogeneous system?

A nonhomogeneous linear system has many solutions that can be expressed in parametric vector form. Signup and view all the answers

How can we describe a solution set of Ax = b geometrically?

As a translation where vector addition moves v by p to v + p. Signup and view all the answers

What is the solution set of Ax = b?

A line through p parallel to the solution set of Ax = 0. Signup and view all the answers

Match Theorem 6 components with their descriptions:

Stationary solution = Assume Ax=b is consistent for some given b, and let p be a solution General solution = The set of all vectors of the form w = p + v(h), where v(h) is any solution of Ax=0 Nonzero solution requirement = Only applies to an equation Ax=b that has at least one nonzero solution p Signup and view all the answers

What are the steps to write a solution set in parametric vector form?

Row reduce the augmented matrix to reduced echelon form and express each basic variable in terms of free variables. Signup and view all the answers

Study Notes

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.

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Description

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.