Questions and Answers
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.
True
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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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
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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.
