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Questions and Answers
What are the three requirements for echelon form?
What are the three requirements for echelon form?
- All non-zero rows at the bottom
- All entries in column below a pivot are non-zero
- All non-zero rows at the top (correct)
- Each leading entry of a row is in a column to the right of the leading entry of the row above it (correct)
What are the two requirements for a matrix to be row reduced?
What are the two requirements for a matrix to be row reduced?
- Leading entry is zero (correct)
- All entries below leading entries are non-zero
- All rows are zero rows
- Each leading 1 is the only non-zero entry in its column (correct)
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.
False (B)
Can a system of linear equations with fewer equations than unknowns have a unique solution?
Can a system of linear equations with fewer equations than unknowns have a unique solution?
What is the geometric relationship between vectors u, -v, and u-v?
What is the geometric relationship between vectors u, -v, and u-v?
Span{u, v} contains only the line through u and the line through v and the origin.
Span{u, v} contains only the line through u and the line through v and the origin.
What does the homogeneous equation Ax=0 imply?
What does the homogeneous equation Ax=0 imply?
When does a nontrivial solution exist?
When does a nontrivial solution exist?
What do we 'say' when vector p is added to vector v?
What do we 'say' when vector p is added to vector v?
What are the four steps in writing a solution set in parametric vector form?
What are the four steps in writing a solution set in parametric vector form?
When is a matrix considered linearly independent?
When is a matrix considered linearly independent?
When are a set of 2 vectors linearly dependent?
When are a set of 2 vectors linearly dependent?
If a set contains more entries in each vector, what can be said about the set?
If a set contains more entries in each vector, what can be said about the set?
If a set contains the zero vector, then what can be said about the set?
If a set contains the zero vector, then what can be said about the set?
A homogeneous equation is always consistent.
A homogeneous equation is always consistent.
The equation Ax=0 gives an explicit description of its solution set.
The equation Ax=0 gives an explicit description of its solution set.
The solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the equation Ax=0.
The solution set of Ax=b is the set of all vectors of the form w=p+vh, where vh is any solution of the equation Ax=0.
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Study Notes
Echelon and Reduced Row Echelon Form
- Echelon form requires that all non-zero rows are at the top, leading entries are shifted right compared to the row above, and all entries below a pivot are zero.
- Row-reduced form requires leading entries to be 1, which must be the only non-zero entry in their columns, and can have zero rows.
Parametric Description and Solutions
- Finding a parametric description of a linear system’s solution set is only valid if the system has at least one solution; otherwise, the statement is false.
- An underdetermined system, having fewer equations than unknowns, cannot have a unique solution due to at least one free variable in the system.
Geometric Relationships
- Vectors ( u, -v, ) and ( u-v ) form a parallelogram with one vertex at the origin (0).
- Span of vectors ( {u, v} ) includes all linear combinations, not just the lines through ( u ) and ( v ).
Homogeneous Systems
- A homogeneous equation of the form ( Ax=0 ) always has at least one solution, the trivial solution.
- Nontrivial solutions in homogeneous equations exist when there is at least one free variable present.
Vector Translation
- Translating vector ( v ) by point ( p ) results in the vector ( v + p ), effectively moving ( v ) in a direction parallel to the line defined by ( p ) to the origin.
Writing Solution Sets
- To write a solution set in parametric vector form, follow these steps: row reduce the matrix, express basic variables in terms of free variables, write the typical solution as a vector, and decompose this vector into a linear combination of vectors using free variables.
Linear Independence and Dependence
- A matrix is deemed linearly independent if only the trivial solution exists, indicating no vector can be expressed as a combination of others.
- A set of two vectors is linearly dependent if one is a scalar multiple of the other.
- A set is inherently linearly dependent if it contains more vectors than dimensions (more entries in each vector) or includes the zero vector.
Consistency of Homogeneous Equations
- Homogeneous equations are always consistent, meaning they have at least the trivial solution.
- The equation ( Ax=0 ) provides an implicit description of the solution set; explicit solutions are found through solving the equation.
Solution Sets of Non-Homogeneous Equations
- The solution set of ( Ax=b ) may be empty; it is only valid under the condition that the equation is consistent for a particular ( b ) and there exists a specific solution vector ( p ).
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