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Questions and Answers
Every matrix is row equivalent to a unique matrix in echelon form.
Every matrix is row equivalent to a unique matrix in echelon form.
False
Every matrix is row equivalent to a unique matrix in reduced row echelon form.
Every matrix is row equivalent to a unique matrix in reduced row echelon form.
True
Any system of n linear equations in n variables has at most n solutions.
Any system of n linear equations in n variables has at most n solutions.
False
If a system of linear equations has two different solutions, it must have an infinite number of solutions.
If a system of linear equations has two different solutions, it must have an infinite number of solutions.
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If a system has no free variables, it has a single solution.
If a system has no free variables, it has a single solution.
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If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have the same solutions.
If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have the same solutions.
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The equation Ax = 0 has the trivial solution if and only if there are no free variables.
The equation Ax = 0 has the trivial solution if and only if there are no free variables.
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If A is an m x n matrix and the equation Ax = b is consistent for every b in R^m, then A has m pivot columns.
If A is an m x n matrix and the equation Ax = b is consistent for every b in R^m, then A has m pivot columns.
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If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.
If an n x n matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.
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If Set U, V, W is linearly independent, then u, v, and w are not in R^2.
If Set U, V, W is linearly independent, then u, v, and w are not in R^2.
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If u and v are in R^m, then -u is in Span(u, v).
If u and v are in R^m, then -u is in Span(u, v).
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If a system Ax = b has more than one solution, then so does the system Ax = 0.
If a system Ax = b has more than one solution, then so does the system Ax = 0.
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If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.
If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.
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If matrices A and B are row equivalent, then they have the same reduced echelon form.
If matrices A and B are row equivalent, then they have the same reduced echelon form.
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If an m x n matrix A has a pivot position in every row, then the equation Ax has a unique solution for each b in R^m.
If an m x n matrix A has a pivot position in every row, then the equation Ax has a unique solution for each b in R^m.
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If 3x3 matrices A and B each have 3 pivot positions, then A can be transformed into B by elementary row operations.
If 3x3 matrices A and B each have 3 pivot positions, then A can be transformed into B by elementary row operations.
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Study Notes
Echelon Forms
- Every matrix is not row equivalent to a unique matrix in row echelon form; different echelon forms can exist due to row multiplication by nonzero constants.
- Every matrix is row equivalent to a unique matrix in reduced row echelon form.
Linear Equations and Solutions
- A system of ( n ) linear equations in ( n ) variables can have an infinite number of solutions, not just a maximum of ( n ).
- If a linear system has two different solutions, it indicates the presence of at least one free variable, leading to infinitely many solutions.
- A system without free variables can have at most one solution, but it may not guarantee a single solution.
Augmented Matrices and Row Operations
- Transforming an augmented matrix ([A \ b]) to ([C \ d]) using elementary row operations keeps the solution set unchanged for systems ( Ax = b ) and ( Cx = d ).
- The homogeneous equation ( Ax = 0 ) always possesses the trivial solution, independent of the presence of free variables.
Matrix Properties and Theorems
- If an ( m \times n ) matrix ( A ) allows the equation ( Ax = b ) to be consistent for every ( b \in R^m ), then ( A ) must have ( m ) pivot columns.
- A square ( n \times n ) matrix ( A ) possessing ( n ) pivot positions ensures its reduced echelon form is the identity matrix.
- An ( r )-dimensional set composed of 3 vectors in ( R^2 ) cannot be linearly independent due to the limitation imposed by dimension.
Span and Linear Combinations
- The vector (-u) belongs to ( \text{Span}(u, v) ) since it can be expressed as a linear combination: (-u = -1u + 0v).
- A system that has multiple solutions for ( Ax = b ) implies that the associated homogeneous system ( Ax = 0 ) will also have multiple solutions.
Consistency and Reduced Echelon Form
- An augmented matrix ([A \ b]) achieving reduced echelon form does not guarantee that ( Ax = b ) is consistent.
- Row equivalent matrices ( A ) and ( B ) share the same reduced echelon form.
Unique Solutions and Pivot Positions
- An ( m \times n ) matrix ( A ) containing a pivot position in every row does not automatically imply that ( Ax ) has a unique solution for each ( b \in R^m ).
- If two ( 3 \times 3 ) matrices, ( A ) and ( B ), each feature three pivot positions, then they can be connected through elementary row operations.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge on Linear Algebra concepts from Chapters 1 & 2 through this interactive flashcard quiz. Explore the uniqueness of echelon forms and the solutions of linear equations as you challenge yourself with these statements.
