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?
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The homogeneous equation Ax=0 has a nontrivial solution if it has at least one free variable.
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A nontrivial solution x can only consist of nonzero entries.
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What is the ONLY solution in Span{0}?
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If the equation Ax=0 has only one free variable, then the solution set is a line through the origin.
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What is a parametric vector equation?
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Whenever a solution is described explicitly with vectors, we say that the solution is in parametric vector form.
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What characterizes a nonhomogeneous system?
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How can we describe a solution set of Ax = b geometrically?
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What is the solution set of Ax = b?
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Match Theorem 6 components with their descriptions:
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What are the steps to write a solution set in parametric vector form?
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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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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.
