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Questions and Answers
A linear transformation is a special type of function.
A linear transformation is a special type of function.
True
If A is an m x n matrix, then the range of the transformation x to Ax is Rm.
If A is an m x n matrix, then the range of the transformation x to Ax is Rm.
False
If A is a 3x5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R3.
If A is a 3x5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R3.
False
Every linear transformation is a matrix transformation.
Every linear transformation is a matrix transformation.
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A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.
A transformation T is linear if and only if T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all v1 and v2 in the domain of T and for all scalars c1 and c2.
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Every matrix transformation is a linear transformation.
Every matrix transformation is a linear transformation.
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The codomain of the transformation x to Ax is the set of all linear combinations of the columns of A.
The codomain of the transformation x to Ax is the set of all linear combinations of the columns of A.
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A linear transformation preserves the operations of vector addition and scalar multiplication.
A linear transformation preserves the operations of vector addition and scalar multiplication.
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The superposition principle is a physical description of a linear transformation.
The superposition principle is a physical description of a linear transformation.
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What is the domain of a transformation?
What is the domain of a transformation?
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What is the range of a transformation?
What is the range of a transformation?
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Two characteristics of a linear transformation are: T(u + v) = ______ and T(cv) = ______.
Two characteristics of a linear transformation are: T(u + v) = ______ and T(cv) = ______.
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In an mxn matrix, what do m and n represent?
In an mxn matrix, what do m and n represent?
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Every linear transformation maps the zero vector into the zero vector.
Every linear transformation maps the zero vector into the zero vector.
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Study Notes
Linear Transformations
- A linear transformation is a function between sets of vectors, preserving structure.
- For a transformation T defined by T(x) = Ax with an m x n matrix A, the range is the set of linear combinations of A's columns, not R^m.
Matrix Transformations
- Every matrix transformation qualifies as a linear transformation, but not all linear transformations can be expressed as matrix transformations.
- The domain of a transformation T is R^n, correlating with the number of columns in matrix A.
Properties of Linear Transformations
- A transformation T is linear if it satisfies:
- T(u + v) = T(u) + T(v) for all u, v in the domain.
- T(cv) = cT(v) for any scalar c and vector v in the domain.
- Linear transformations retain vector addition and scalar multiplication operations (superposition principle).
Definitions
- The domain of a transformation comprises all vectors x for which T(x) is defined.
- The range is the collection of vectors generated as T(x) for x in the domain.
Key Concepts
- The transformation maps the zero vector to the zero vector, confirming linearity.
- The codomain of T is characterized as the set of all linear combinations of A's columns, which reflects the range's definition.
- Two essential characteristics of linear transformations include the additive and homogeneity properties.
Matrix Dimensions
- For an m x n matrix A, 'm' represents the range dimension while 'n' indicates the domain dimension.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of linear transformations and their properties with these flashcards from Linear Algebra Chapter 1.8. This quiz covers key concepts including the definition of linear transformations and the range of transformations defined by matrices.
