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Every elementary row operation is reversible.
Every elementary row operation is reversible.
True
A 5x6 matrix has six rows.
A 5x6 matrix has six rows.
False
The solution set of a linear system involving variables x1,...xn is a list of numbers (s1,...sn) that makes each equation in the system a true statement when the values s1,...sn are substituted for x1,...xn respectively.
The solution set of a linear system involving variables x1,...xn is a list of numbers (s1,...sn) that makes each equation in the system a true statement when the values s1,...sn are substituted for x1,...xn respectively.
False
Two fundamental questions about a linear system involve existence and uniqueness.
Two fundamental questions about a linear system involve existence and uniqueness.
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Two matrices are row equivalent if they have the same number of rows.
Two matrices are row equivalent if they have the same number of rows.
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Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
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Two equivalent linear systems can have different solution sets.
Two equivalent linear systems can have different solution sets.
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A consistent system of linear equations has one or more solutions.
A consistent system of linear equations has one or more solutions.
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In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
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The row reduction algorithm applies only to augmented matrices for a linear system.
The row reduction algorithm applies only to augmented matrices for a linear system.
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A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
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Finding a parametric description of the solution set of a linear system is the same as solving the system.
Finding a parametric description of the solution set of a linear system is the same as solving the system.
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If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.
If one row in an echelon form of an augmented matrix is [0 0 0 5 0], then the associated linear system is inconsistent.
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The reduced echelon form of a matrix is unique.
The reduced echelon form of a matrix is unique.
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If every column of an augmented matrix contains a pivot, then the corresponding system is consistent.
If every column of an augmented matrix contains a pivot, then the corresponding system is consistent.
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The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
The pivot positions in a matrix depend on whether row interchanges are used in the row reduction process.
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A general solution of a system is an explicit description of all solutions of the system.
A general solution of a system is an explicit description of all solutions of the system.
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Whenever a system has free variables, the solution set contains many solutions.
Whenever a system has free variables, the solution set contains many solutions.
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Another notation for the vector [-4 over 3] is [-4 3].
Another notation for the vector [-4 over 3] is [-4 3].
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The points in the plane corresponding to [-2 5] and [-5 2] lie on a line through the origin.
The points in the plane corresponding to [-2 5] and [-5 2] lie on a line through the origin.
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An example of a linear combination of vectors v1 and v2 is the vector 1/2v1.
An example of a linear combination of vectors v1 and v2 is the vector 1/2v1.
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The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b.
The solution set of the linear system whose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b.
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The set span {u,v} is always visualized as a plane through the origin.
The set span {u,v} is always visualized as a plane through the origin.
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When u and v are nonzero vectors, span {u,v} contains only the line through u and the origin, and the line through v and the origin.
When u and v are nonzero vectors, span {u,v} contains only the line through u and the origin, and the line through v and the origin.
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Any list of five real numbers is a vector in R5.
Any list of five real numbers is a vector in R5.
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Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in span {a1, a2, a3}.
Asking whether the linear system corresponding to an augmented matrix [a1 a2 a3 b] has a solution amounts to asking whether b is in span {a1, a2, a3}.
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The vector v results when a vector u-v is added to the vector v.
The vector v results when a vector u-v is added to the vector v.
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The weights c1...cp in a linear combination c1v1 +...+ cpvp cannot all be zero.
The weights c1...cp in a linear combination c1v1 +...+ cpvp cannot all be zero.
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The equation Ax=b is referred to as a vector equation.
The equation Ax=b is referred to as a vector equation.
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A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.
A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax=b has at least one solution.
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Study Notes
Elementary Row Operations
- Every elementary row operation is reversible.
Matrix Dimensions
- A 5x6 matrix has 5 rows and 6 columns.
Solution Sets
- The solution set of a linear system consists of all possible solutions, not just a single list of values.
Fundamental Questions
- Two essential questions regarding linear systems are existence and uniqueness of solutions.
Row Equivalence
- Two matrices are row equivalent if one can be transformed into the other through a series of row operations, regardless of the number of rows.
Augmented Matrices
- Elementary row operations on an augmented matrix do not alter the solution set of the associated linear system.
Equivalent Systems
- Two equivalent linear systems will always share the same solution set.
Consistent Systems
- A consistent system of equations will have at least one solution.
Reduced Echelon Form
- The reduced echelon form of a matrix is unique, meaning any matrix can only be row-reduced to one consistent form.
Row Reduction Algorithm
- The row reduction algorithm can be applied to any matrix, not limited to just augmented matrices.
Basic Variables
- Basic variables in a linear system correspond directly to pivot columns found in the coefficient matrix.
Parametric Descriptions
- Finding a parametric description of a solution set for a linear system equates to solving the system itself.
Inconsistent Systems
- A row of the form [0 0 0 5 0] in an echelon form indicates an inconsistent system, as it suggests a false equation.
Pivot Positions
- The locations of pivot positions do not rely on the presence of row interchanges during the row reduction process.
General Solutions
- A general solution provides an explicit description encompassing all potential solutions to a system.
Free Variables
- The existence of free variables in a system does not guarantee multiple solutions; an inconsistent system will have no solutions.
Vector Representation
- The notation for a vector cannot vary; [-4 over 3] is not the same as [-4 3].
Points on a Line
- The points corresponding to vectors [-2 5] and [-5 2] do not lie on a line through the origin.
Linear Combinations
- A linear combination like 1/2v1 is valid if v1 and v2 belong to Rn.
Solution Set Relationships
- The solution set of the matrix [a1 a2 a3 b] mirrors the solution set of the associated linear equation x1a1 + x2a2 + x3a3 = b.
Span of Vectors
- Span {u,v} is not always a plane through the origin; it can represent a line.
Vector Spaces
- Any selection of five real numbers constitutes a vector in R5.
Solutions in Span
- The question of whether the linear system [a1 a2 a3 b] has solutions is equivalent to asking if vector b resides in the span {a1, a2, a3}.
Vector Operations
- The operation u - v + v results in the vector u, demonstrating how vector addition functions.
Zero Weights
- In a linear combination, the weights c1...cp can indeed all be zero.
Vector Equations
- Ax = b is not solely a vector equation; it depicts a broader algebraic relationship.
Linear Combinations and Solutions
- A vector b can be expressed as a linear combination of the columns of matrix A if and only if the equation Ax = b holds true with at least one solution.
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Description
Test your understanding of Linear Algebra concepts with these True/False flashcards. Each flashcard presents a statement related to row operations, matrix dimensions, and solution sets. Challenge yourself and strengthen your knowledge of the subject!
