Questions and Answers
A linear transformation T: R^n -> R^m is completely determined by its effects on the columns of the n x n identity matrix.
True
If T: R^2 -> R^2 rotates vectors about the origin through an angle theta, then T is a linear transformation.
True
When two linear transformations are performed one after another, the combined effect may not always be a linear transformation.
False
A mapping T: R^n -> R^m is onto R^m if every vector x in R^n maps onto some vector in R^m.
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If A is a 3x2 matrix, then the transformation x -> Ax cannot be one to one.
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If A is a 4x3 matrix, then the transformation x -> Ax maps R^3 onto R^4.
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Every linear transformation from R^n to R^m is a matrix transformation.
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What is a linear system?
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A system of linear equations can have how many solutions?
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A consistent system of linear equations has one or infinitely many solutions.
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An inconsistent system of linear equations has no solution.
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What does it mean for a linear system to be consistent?
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What does it mean for a linear system to be inconsistent?
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What are elementary row operations?
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Every elementary row operation is reversible.
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A 5x6 matrix has 6 rows.
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The solution set of a linear system with variables $x_1...x_n$ consists of a list of numbers that satisfy the equations.
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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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What is a leading entry in a matrix?
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What defines row echelon form?
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What is reduced row echelon form?
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Homogeneous equations can be inconsistent.
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The trivial solution exists in any homogeneous equation Ax=0.
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A set can be linearly dependent if it contains the zero vector.
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If a set contains more vectors than there are entries in a single vector, then the set is linearly dependent.
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Every matrix transformation is a linear transformation.
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A linear transformation T: R^n -> R^m always maps the origin of R^n to the origin of R^m.
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If T(x) = 0 has only the trivial solution, then T is one-to-one.
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A linear transformation preserves the operations of vector addition and scalar multiplication.
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The equation ax=b is homogeneous if the zero vector is a solution.
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Study Notes
Linear Systems and Solutions
- A linear system consists of one or more linear equations involving the same set of variables.
- Possible outcomes for a system: no solution, exactly one solution, or infinitely many solutions.
- A system is consistent if it has one or infinitely many solutions; it's inconsistent if it has no solutions.
Elementary Row Operations
- Replacement: Replace one row by itself plus a multiple of another row.
- Interchange: Swap two rows.
- Scaling: Multiply all entries in a row by a non-zero constant.
- Elementary row operations are reversible.
Matrix and Solution Properties
- A 5x6 matrix has 5 rows and 6 columns.
- The solution set of a linear system can consist of all possible values that satisfy each equation, not just a single list of numbers.
- Two matrices are row equivalent if one can be transformed into the other through a series of row operations.
Echelon Forms
- Row Echelon Form: All non-zero rows are above any rows of zeros; the leading entry of each subsequent row is in the column to the right of the leading entry above it.
- Reduced Row Echelon Form: Each leading entry is 1 and is the only non-zero entry in its column.
Theorems and Consistency
- Each matrix has a unique reduced echelon form (Theorem 1).
- A system is consistent if the rightmost column of its augmented matrix is not a pivot column (Theorem 2).
- All valid linear systems that are equivalent share the same solution set (Theorem 4).
Variables and Solutions
- A basic variable corresponds to a pivot column of the coefficient matrix.
- Finding a parametric description of a solution set is equivalent to solving the system.
- Inconsistent systems may contain rows like ([0 , 0 , 0 , 5 , 0]), which do not lead to contradictions.
Linear Dependence and Independence
- A set of vectors is linearly dependent if one vector can be expressed as a combination of others, especially when there are more vectors than entries.
- A set of vectors is linearly independent if no vector can be represented as a linear combination of others.
- The presence of the zero vector in a set also indicates linear dependence.
Transformations and Ranges
- A linear transformation preserves vector addition and scalar multiplication.
- The range of a transformation defined by a matrix is the set of all linear combinations of its columns.
- Matrix transformations are a subclass of linear transformations and every linear transformation can be represented this way.
Additional Theorems
- A transformation ( T ) is one-to-one if ( T(x) = 0 ) has only the trivial solution (Theorem 11).
- A linear transformation defined by a matrix spans ( R^m ) if ( Ax = b ) is consistent for every ( b \in R^m ) (Theorem 12).
- A linear transformation can be explicitly determined by its action on the identity matrix (Theorem 10).
Miscellaneous
- The equation ( Ax=b ) is always consistent if the zero vector is a solution to the corresponding homogeneous equation ( Ax=0 ).
- Every linear transformation from ( R^n ) to ( R^m ) can be described by a matrix transformation, reinforcing the connection between linear algebra and matrix theory.
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Description
This quiz focuses on linear systems, their solutions, and elementary row operations. Explore concepts such as consistency, matrix properties, and echelon forms. Test your understanding of the foundational principles of linear algebra.
