• Title: Examples of Abstract Vector Spaces

  • Series: Abstract Linear Algebra

  • YouTube-Title: Abstract Linear Algebra 2 | Examples of Abstract Vector Spaces

  • Bright video: https://youtu.be/ZA4rGQ2oGnM

  • Dark video: https://youtu.be/ppJE8Y1cNlI

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ala02_sub_eng.srt missing

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  • Quiz Content

    Q1: Consider the real vector space $\mathcal{P}(\mathbb{R})$ given by the polynomials. What is the vector addition of $p_1(x) = x^2$ and $p_2(x) = -(x-1)^2$?

    A1: $(p_1 + p_2)(x) = 2x -1$

    A2: $(p_1 + p_2)(x) = 2x + 1$

    A3: $(p_1 + p_2)(x) = x^2 + 2x -1$

    A4: $(p_1 + p_2)(x) = x^2 + 2x + 1$

    A5: It’s not defined.

    Q2: Consider the real vector space $\mathcal{P}(\mathbb{R})$ given by the polynomials. Is the element $x \mapsto \sum_{k = 1}^n \frac{x^k}{k!}$ an element of $\mathcal{P}(\mathbb{R})$ for any given $n \in \mathbb{N}$?

    A1: Yes, it is.

    A2: No, it is not a polynomial.

    A3: No, it’s not a well-defined function.

    A4: One needs more information.

    Q3: Consider the real vector space $\mathcal{P}(\mathbb{R})$ given by the polynomials. Is the element $x \mapsto \sum_{k = 1}^\infty \frac{x^k}{k!}$ an element of $\mathcal{P}(\mathbb{R})$?

    A1: Yes, it is.

    A2: No, it is not a polynomial.

    A3: No, it’s not a well-defined function.

    A4: One needs more information.

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