Questions and Answers
What is a homogeneous linear system?
A homogeneous linear system always has at least one solution.
True
What is the trivial solution?
The zero solution.
What is a nontrivial solution?
Signup and view all the answers
The homogeneous equation Ax=0 has a nontrivial solution if it has at least one free variable.
Signup and view all the answers
A nontrivial solution x can only consist of nonzero entries.
Signup and view all the answers
What is the ONLY solution in Span{0}?
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.
Signup and view all the answers
What is a parametric vector equation?
Signup and view all the answers
Whenever a solution is described explicitly with vectors, we say that the solution is in parametric vector form.
Signup and view all the answers
What characterizes a nonhomogeneous system?
Signup and view all the answers
How can we describe a solution set of Ax = b geometrically?
Signup and view all the answers
What is the solution set of Ax = b?
Signup and view all the answers
Match Theorem 6 components with their descriptions:
Signup and view all the answers
What are the steps to write a solution set in parametric vector form?
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
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.