Podcast
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 a 3×5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is ℝ3.
If A is a 3×5 matrix and T is a transformation defined by T(x)=Ax, then the domain of T is ℝ3.
False
If A is an m×n matrix, then the range of the transformation x maps to Ax is ℝ^m.
If A is an m×n matrix, then the range of the transformation x maps to Ax is ℝ^m.
False
Every linear transformation is a matrix transformation.
Every linear transformation is a matrix transformation.
Signup and view all the answers
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.
Signup and view all the answers
The range of the transformation x Ax is the set of all linear combinations of the columns of A.
The range of the transformation x Ax is the set of all linear combinations of the columns of A.
Signup and view all the answers
Every matrix transformation is a linear transformation.
Every matrix transformation is a linear transformation.
Signup and view all the answers
If T: R^n -> R^m is a linear transformation and if c is in R^m, then a uniqueness question is 'Is c in the range of T?'
If T: R^n -> R^m is a linear transformation and if c is in R^m, then a uniqueness question is 'Is c in the range of T?'
Signup and view all the answers
A linear transformation preserves the operations of vector addition and scalar multiplication.
A linear transformation preserves the operations of vector addition and scalar multiplication.
Signup and view all the answers
A linear transformation T: R^n -> R^m always maps the origin of R^n to the origin of R^m.
A linear transformation T: R^n -> R^m always maps the origin of R^n to the origin of R^m.
Signup and view all the answers
Study Notes
Linear Transformations
- A linear transformation maps vectors from R^n to ℝ^m, assigning each vector x in R^n to vector T(x) in ℝ^m.
- For a transformation defined by T(x) = Ax, if A is a 3×5 matrix, the domain of T is R^5, not R^3. The input vector x must match the second dimension of matrix A.
Range and Domain
- The range of the transformation x ↦ Ax is not necessarily ℝ^m; instead, it consists of all linear combinations of the columns of matrix A.
- A matrix transformation is a specific type of linear transformation represented as x ↦ Ax, establishing that not all linear transformations are matrix transformations.
Properties of Linear Transformations
- A transformation T is linear if it meets the condition T(c1v1 + c2v2) = c1T(v1) + c2T(v2) for all vectors v1, v2, and scalars c1, c2 in the domain.
- Linear transformations preserve vector addition and scalar multiplication, demonstrated by T(cu + dv) = cT(u) + dT(v).
Specific Characteristics
- The range of the transformation x ↦ Ax includes all linear combinations of the columns of A, confirming a link between the transformation and the structure of matrix A.
- Every matrix transformation is indeed a linear transformation, following the properties of linearity in T.
Existence and Uniqueness Questions
- In context of linear transformations T: R^n → R^m, questioning if a vector c is in the range of T pertains to existence (whether there is an x such that T(x) = c) rather than uniqueness.
Origin Mapping
- A linear transformation T: R^n → R^m guarantees that the origin of R^n (T(0)) is always mapped to the origin of R^m (0), affirming the structure-preserving nature of linear transformations.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of linear transformations with this interactive flashcard quiz. Explore key concepts and definitions that are essential to mastering this part of linear algebra.
