Questions and Answers
Every elementary row operation is reversible.
True
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.
False
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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].
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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.
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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.
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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.
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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}.
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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.
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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.
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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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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!
