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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 variables has at most n solutions.
Any system of n linear equations in n variables 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 mxn matrix and the equation Ax=b is consistent for some b, then the columns of A span R^m.
If A is an mxn 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 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.
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.
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 mxn matrix and the equation Ax=b is consistent for every b in R^m, then A has m pivot columns.
If A is an mxn matrix and the equation Ax=b is consistent for every b in R^m, then A has m pivot columns.
If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a unique solution for each b in R^m.
If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a unique solution for each b in R^m.
If an nxn matrix A has n pivot positions, then the reduced echelon form of A is the nxn identity matrix.
If an nxn matrix A has n pivot positions, then the reduced echelon form of A is the nxn identity matrix.
If 3x3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
If 3x3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
If A is an mxn matrix, if the equation Ax=b has at least two different solutions, and if the equation Ax=c is consistent, then Ax=c has many solutions.
If A is an mxn matrix, if the equation Ax=b has at least two different solutions, and if the equation Ax=c is consistent, then Ax=c has many solutions.
If A and B are row equivalent mxn matrices and if the columns of A span R^m, then so do the columns of B.
If A and B are row equivalent mxn 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^m.
In some cases, it is possible for four vectors to span R^m.
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, and w are nonzero vectors in R^2, then w is a linear combination of u and v.
If u,v, and 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 T:R^n, then u is a linear combination of v and w.
If w is a linear combination of u and v in T:R^n, then u is a linear combination of v and w.
Suppose that v1,v2, and 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 dependent.
Suppose that v1,v2, and 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 dependent.
A linear transformation is a function.
A linear transformation is a function.
If A is a 6x5 matrix, the linear transformation x->Ax cannot map R^5 onto R^6.
If A is a 6x5 matrix, the linear transformation x->Ax cannot map R^5 onto R^6.
If A is an mxn matrix with m pivot columns, then the linear transformation x->Ax is a one-to-one mapping.
If A is an mxn 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 Solutions
- Every matrix is row equivalent to a unique matrix in reduced row-echelon form (RREF), which is distinct for each matrix.
- A system of n linear equations in n variables can have either unique, infinite, or no solutions, not limited to at most n solutions.
- If a system has two different solutions, it must have infinitely many due to the nature of linear equations.
- A system without free variables should still be examined carefully, as it may not guarantee a unique solution.
Augmented Matrices and Consistency
- Transformed augmented matrices [A b] and [C d] maintain equivalent solution sets when obtained through elementary row operations.
- If a system Ax=b has more than one solution, the corresponding homogeneous system Ax=0 also has or continues to have multiple solutions.
- With an mxn matrix A, if Ax=b is consistent for some b, it does not imply that A spans R^m.
Echelon Forms and Pivot Columns
- Consistency in the equation Ax=b is uncertain even if an augmented matrix can be transformed to reduced echelon form.
- Row equivalent matrices A and B have the same reduced echelon form.
- In an mxn matrix A, there must be m pivot columns if Ax=b is consistent for every b in R^m.
- A matrix A with a pivot position in every row ensures unique solutions for Ax=b for each b in R^m.
Linear Independence and Transformations
- An nxn matrix with n pivot positions does not necessarily reduce to the identity matrix.
- If 3x3 matrices A and B have pivot positions in all rows, they can be transformed into each other using elementary row operations.
- A matrix A can lead to multiple solutions in Ax=c if it already has multiple solutions for Ax=b.
Span and Linear Combinations
- For a set S={v1,v2,v3} in R^3, not being scalar multiples does not guarantee linear independence.
- A linearly independent set {u,v,w} cannot reside in R^2 but must be in R^3 or higher.
- The possibility of four vectors spanning R^m is not guaranteed.
Properties of Vector Transformations
- Any vector -u, derived from vector u in R^m, can be included in the span of {u,v}.
- Nonzero vectors u, v, and w in R^2 may not guarantee that w is a linear combination of u and v.
- In R^n, if w is a linear combination of u and v, it does not imply that u is expressible as a combination of v and w.
Linear Transformations and Mapping
- A linear transformation is fundamentally a function mapping vectors in one vector space to another.
- A linear transformation from a 6x5 matrix cannot cover R^6 completely due to dimensionality constraints.
- An mxn matrix A with m pivot columns will not ensure a one-to-one mapping unless each column also has a pivot, necessitating at least as many pivot columns as total columns.
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