Stanford ENGR108: Introduction to Applied Linear Algebra | 2020 | Lecture 7 - VMLS linear functions | Summary and Q&A

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February 24, 2021
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Stanford ENGR108: Introduction to Applied Linear Algebra | 2020 | Lecture 7 - VMLS linear functions

TL;DR

Linear and affine functions are mathematical concepts that involve mapping vectors to real numbers, with linear functions satisfying the superposition property and affine functions adding a constant offset.

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Questions & Answers

Q: What does the notation for linear functions represent?

The notation f: R^n -> R indicates that f is a function that takes an n-dimensional vector as input and returns a real number as output.

Q: What is the superposition property for linear functions?

The superposition property states that for linear functions, the order of forming linear combinations and applying the function does not matter. This property allows for flexibility in mathematical operations.

Q: Are all linear functions also affine functions?

Yes, all linear functions are affine functions because they can be represented as the sum of a linear term and a constant term.

Q: How is an affine function different from a linear function?

An affine function is a linear function with an additional constant offset. This offset can shift the entire function without affecting its linearity.

Q: Can affine functions have negative weights in linear combinations?

Yes, affine functions can have negative weights as long as the sum of the weights in a linear combination adds up to one. This property is known as the superposition property for affine functions.

Summary & Key Takeaways

  • Linear functions take n numbers as input and return a real number as output.

  • The superposition property states that linear functions commute with forming linear combinations, allowing for flexibility in the order of operations.

  • Affine functions are a combination of linear functions and a constant offset.

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