Questions and Answers
What is a fundamental property of a linear transformation?
What is the representation of a linear transformation?
What is a characteristic of an injection?
What is a result of the composition of two linear transformations?
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What is a common application of linear transformations in data science?
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What is the purpose of dimensionality reduction?
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What is true about the inverse of a linear transformation?
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What is a result of the scalar multiplication of a linear transformation?
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Study Notes
Linear Transformations
Definition
A linear transformation is a function between vector spaces that preserves the operations of vector addition and scalar multiplication.
Properties

Linearity: A function T: V → W is linear if it satisfies:
 T(av) = aT(v) for any scalar a and vector v
 T(v + w) = T(v) + T(w) for any vectors v and w
 Additivity: T(v + w) = T(v) + T(w)
 Homogeneity: T(av) = aT(v)
Matrix Representation
 A linear transformation can be represented by a matrix
 If T: ℝⁿ → ℝᵐ is a linear transformation, then there exists an m x n matrix A such that:
 T(v) = Av for any vector v in ℝⁿ
Operations on Linear Transformations
 Composition: The composition of two linear transformations is also a linear transformation
 Inverse: If a linear transformation has an inverse, it is also a linear transformation
 Scalar Multiplication: The scalar multiple of a linear transformation is also a linear transformation
Types of Linear Transformations
 Injection (OnetoOne): A linear transformation that maps distinct inputs to distinct outputs
 Surjection (Onto): A linear transformation that maps to every output in the codomain
 Bijection (OnetoOne Correspondence): A linear transformation that is both an injection and a surjection
Applications in Data Science
 Data Transformation: Linear transformations can be used to transform data from one space to another
 Dimensionality Reduction: Linear transformations can be used to reduce the dimensionality of highdimensional data
 Feature Extraction: Linear transformations can be used to extract relevant features from data
Linear Transformations
Definition and Properties
 A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication.
 Properties of linear transformations include:
 Linearity: T(av) = aT(v) and T(v + w) = T(v) + T(w)
 Additivity: T(v + w) = T(v) + T(w)
 Homogeneity: T(av) = aT(v)
Matrix Representation
 A linear transformation can be represented by a matrix A, where T(v) = Av for any vector v.
Operations on Linear Transformations
 Composition of two linear transformations is also a linear transformation.
 Inverse of a linear transformation, if it exists, is also a linear transformation.
 Scalar multiple of a linear transformation is also a linear transformation.
Types of Linear Transformations
 Injection (OnetoOne): Maps distinct inputs to distinct outputs.
 Surjection (Onto): Maps to every output in the codomain.
 Bijection (OnetoOne Correspondence): Both an injection and a surjection.
Applications in Data Science
 Data transformation: Linear transformations can be used to transform data from one space to another.
 Dimensionality reduction: Linear transformations can reduce the dimensionality of highdimensional data.
 Feature extraction: Linear transformations can extract relevant features from data.
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Description
Learn about linear transformations, their properties, and matrix representation in vector spaces. Understand linearity, additivity, and homogeneity of linear transformations.