• Title: Linear Maps (Definition)

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 18 | Linear Maps (Definition)

  • Bright video: https://youtu.be/Xi-HknrH6OM

  • Dark video: https://youtu.be/lJnwwbwPE-8

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

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  • Subtitle on GitHub: la18_sub_eng.srt missing

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  • Quiz Content

    Q1: What is not a property of a linear map $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$?

    A1: $f( \mathbf{x} \cdot \mathbf{y} ) = f( \mathbf{x} ) \cdot f( \mathbf{y} ) $

    A2: $f( \mathbf{x} + \mathbf{y} ) = f( \mathbf{x} ) + f( \mathbf{y} ) $

    A3: $f( \lambda \mathbf{x}) = \lambda f( \mathbf{x} )$

    A4: $f( \mathbf{x} + \lambda \mathbf{y} ) = f( \mathbf{x} ) + \lambda f( \mathbf{y} ) $

    Q2: Let consider the vector space $X = \mathbb{R}$ and a map $f : X \rightarrow X$. Which of the statements is not correct?

    A1: $f(x) = x^2$ is linear.

    A2: $f(x) = x$ is linear.

    A3: $f(x) = 2 x$ is linear.

    A4: $f(x) = 3 x$ is linear.

    A5: $f(x) = 0$ is linear.

    Q3: Let consider the zero vector space $X = {0}$ and a map $f : X \rightarrow X$. Which of the statements is not correct?

    A1: There is a map $f : X \rightarrow X$ that is not linear.

    A2: $f(x) = x^2$ is linear.

    A3: $f(x) = x$ is linear.

    A4: $f(x) = 3 x$ is linear.

    A5: $f(x) = 0$ is linear.

  • Last update: 2024-10

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