Podcast
Questions and Answers
What is the domain of the function T: R^m → R^n?
What is the domain of the function T: R^m → R^n?
- R^n
- R^m (correct)
- Image
- T
What is the codomain 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?
Which of the following statements is true concerning the function T?
- The range of T is a subset of the codomain. (correct)
- Every image of a vector in R^m is in the codomain.
- The range of T is a subset of the domain.
- The range of T is equal to the codomain.
What defines a linear transformation?
What defines a linear transformation?
What is the criterion for A to be a square matrix?
What is the criterion for A to be a square matrix?
What must be verified to show that T(x) = Ax is a linear transformation?
What must be verified to show that T(x) = Ax is a linear transformation?
What does it mean for a vector w to be in the range of T?
What does it mean for a vector w to be in the range of T?
A transformation T is onto if for every vector w in R^n there exists at most one vector u in R^m.
A transformation T is onto if for every vector w in R^n there exists at most one vector u in R^m.
What implies a transformation T is one-to-one?
What implies a transformation T is one-to-one?
What is the trivial solution to the equation T(x) = 0?
What is the trivial solution to the equation T(x) = 0?
When is T one-to-one concerning the columns of the matrix A?
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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