Questions and Answers
What is the domain of the function T: R^m → R^n?
What is the codomain of the function T: R^m → R^n?
R^n
Which of the following statements is true concerning the function T?
What defines a linear transformation?
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What is the criterion for A to be a square matrix?
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What must be verified to show that T(x) = Ax is a linear transformation?
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What does it mean for a vector w to be in the range of T?
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A transformation T is onto if for every vector w in R^n there exists at most one vector u in R^m.
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What implies a transformation T is one-to-one?
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What is the trivial solution to the equation T(x) = 0?
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When is T one-to-one concerning the columns of the matrix A?
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Study Notes
Functions and Transformations
- Domain and codomain define the input and output spaces of a function T: R^m → R^n, with R^m being the domain and R^n the codomain.
- An image is obtained by applying the function T to vectors in the domain, while the range is the set of all such images, represented as range(T), which is a subset of the codomain.
Linear Transformations
- A function T is classified as a linear transformation if it satisfies two conditions:
- The function is additive: T(u + v) = T(u) + T(v) for all u, v in R^m.
- It is homogeneously scalable: T(ru) = rT(u) for all scalars r and vectors u in R^m.
Matrix Properties
- A matrix A has dimensions n x m, signifying n rows and m columns.
- A square matrix occurs when n = m, indicating equal numbers of rows and columns.
Theorem on Linear Transformation
- If T is defined as T(x) = Ax for an n x m matrix A, then T is a linear transformation.
- Verifying linearity involves proving the additivity and scalability conditions hold true for the matrix multiplication.
Characterization of Range
- A vector w is in the range of T if the system Ax = w is consistent, meaning a solution for u exists such that T(u) = w.
- The range(T) is equivalent to the span of the matrix's columns, denoted as span{a1, a2, ..., am}.
One-to-One and Onto Transformations
- A linear transformation T is one-to-one (injective) if each vector in R^n corresponds to at most one vector in R^m.
- T is onto (surjective) if every vector in R^n is the image of at least one vector in R^m.
Trivial Solutions
- A transformation T is one-to-one if the only solution to the equation T(x) = 0 is the trivial solution x = 0.
- For linear transformations, T(0) must equal 0, reinforcing that if T is one-to-one, T(x) = 0 can have only the trivial solution.
Column Independence
- The transformation T defined by T(x) = Ax is one-to-one if and only if the columns of matrix A are linearly independent.
- If matrix A is row-equivalent to matrix B in echelon form, the presence of a pivot in every column of B indicates T is one-to-one.
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Description
Dive into the essential concepts of linear transformations in Chapter 3. This quiz focuses on understanding domains, codomains, images, and ranges essential for comprehending linear mappings. Test your knowledge with key definitions and properties related to these functions.
