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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 (B)
Any system of n linear equations in n variable has at most n solutions.
Any system of n linear equations in n variable has at most n solutions.
False (B)
If a system of linear equations has two different solutions, it must have infinitely many solutions.
If a system of linear equations has two different solutions, it must have infinitely many solutions.
True (A)
If a system of linear equations has no free variables, then it has a unique solution.
If a system of linear equations has no free variables, then it has a unique solution.
If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax=b and Cx=d have exactly the same solution sets.
If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax=b and Cx=d have exactly the same solution sets.
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.
If A is an m x n matrix and the equation Ax=b is consistent for some b, then the columns of A span R^m.
If A is an m x n matrix and the equation Ax=b is consistent for some b, then the columns of A span R^m.
If an augmented matrix [A b] can be transformed by elementary row operations into echelon form, then the equation Ax=b is consistent.
If an augmented matrix [A b] can be transformed by elementary row operations into echelon form, then the equation Ax=b is consistent.
If matrices A and B are row equivalent, they have the same reduced echelon form.
If matrices A and B are row equivalent, they have the same reduced echelon form.
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.
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 positions.
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 positions.
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.
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.
If 3 x 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
If 3 x 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
If A is an m x n matrix, if the equation Ax=b has at least two different solutions, and if the equations Ax=c is consistent, then the equation Ax=c has many solutions.
If A is an m x n matrix, if the equation Ax=b has at least two different solutions, and if the equations Ax=c is consistent, then the equation Ax=c has many solutions.
If A and B are row equivalent m x n matrices and if the columns of A span R^m, then so do the columns of B.
If A and B are row equivalent m x n matrices and if the columns of A span R^m, then so do the columns of B.
If none of the vectors in the set S={v1, v2, v3} in R^3 is a multiple of one of the other vectors, then S is linearly independent.
If none of the vectors in the set S={v1, v2, v3} in R^3 is a multiple of one of the other vectors, then S is linearly independent.
If {u,v,w} is linearly independent, then u,v, and w are not in R^2.
If {u,v,w} is linearly independent, then u,v, and w are not in R^2.
In some cases, it is possible for four vectors to span R^5.
In some cases, it is possible for four vectors to span R^5.
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}.
If u, v, w are nonzero vectors in R^2, then w is a linear combination of u and v.
If u, v, w are nonzero vectors in R^2, then w is a linear combination of u and v.
If w is a linear combination of u and v in R^n, then u is a linear combination of v and w.
If w is a linear combination of u and v in R^n, then u is a linear combination of v and w.
Suppose that v1, v2, v3 are in R^5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1,v2,v3} is linearly independent.
Suppose that v1, v2, v3 are in R^5, v2 is not a multiple of v1, and v3 is not a linear combination of v1 and v2. Then {v1,v2,v3} is linearly independent.
A linear transformation is a function.
A linear transformation is a function.
If A is a 6 x 5 matrix, the linear transformation x |-> Ax cannot map R^5 onto R^6.
If A is a 6 x 5 matrix, the linear transformation x |-> Ax cannot map R^5 onto R^6.
If A is an m x n matrix with m pivot columns, then the linear transformation x |--> Ax is a one-to-one mapping.
If A is an m x n matrix with m pivot columns, then the linear transformation x |--> Ax is a one-to-one mapping.
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Study Notes
Matrix Equivalence and Solution Sets
- Every matrix can be transformed to reduced echelon form, not necessarily a unique matrix in echelon form.
- A system of n linear equations can have infinitely many solutions if the corresponding n×n matrix has fewer than n pivot columns.
- Two distinct solutions in a linear system imply infinitely many solutions exist.
Features of Solution Uniqueness
- A system without free variables guarantees a unique solution; however, it is possible to have no solution even without free variables.
- If an augmented matrix [A b] is modified to [C d] through elementary row operations, both systems share the same solution set.
- More than one solution to Ax=b means Ax=0 must also have multiple solutions.
Column Span and Consistency
- Consistency of Ax=b for one specific vector b does not indicate that the columns of A span R^m.
- A system represented by an augmented matrix can be transformed into echelon form without ensuring that Ax=b is consistent.
- Row equivalent matrices have identical reduced echelon forms.
Pivot Positions and Solutions
- Every n×n matrix with n pivot positions results in its reduced echelon form being the identity matrix.
- An m x n matrix with a pivot in every row guarantees solutions for each b in R^m, but uniqueness cannot be assured if there are fewer pivot columns.
Linear Independence and Span
- A set of vectors in R^3 is linearly independent only if no vector is a combination of others.
- Linear independence of three vectors {u, v, w} implies they cannot exist in R^2.
- A maximum of four vectors cannot span R^5, as five rows require at least five pivot positions.
Combinations and Vector Relationships
- In R^m, -u falls within the span of {u, v}.
- Nonzero vectors u and v in R^2 may not ensure that w lies within their linear combination if they lie on the same line.
- Linear relationships imply that unique combinations do not hold if the vector set is linearly independent.
Linear Transformations and Matrix Properties
- A linear transformation can be defined as a functional mapping.
- A transformation x |-> Ax from a 5-dimensional space cannot map onto a 6-dimensional space if A is a 6x5 matrix.
- For x |-> Ax to be one-to-one, A must have a pivot in every column; a matrix with fewer pivot columns than columns is not one-to-one.
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