• Title: Basis, Linear Independence, Generating Sets

  • Series: Abstract Linear Algebra

  • YouTube-Title: Abstract Linear Algebra 4 | Basis, Linear Independence, Generating Sets

  • Bright video: https://youtu.be/D69BiRT2UDI

  • Dark video: https://youtu.be/7H5526tVkk8

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ala04_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $V = \mathcal{P}(\mathbb{R})$ be the real vector space consisting of polynomials. What is a linear combination of vectors of $V$?

    A1: $x^2 + 5 x^7 - \frac{1}{10} x^{2023}$

    A2: $\sum_{j=1}^\infty x^j$

    A3: $7 x^2 - \sin(x)$

    A4: $x^3 - 6 \frac{1}{x}$

    Q2: Let $V = \mathcal{P}(\mathbb{R})$ be the real vector space consisting of polynomials. What is the span of the set $M$ given by the polynomials $x^2$ and $x^3$?

    A1: $\mathrm{Span}(M) = { \alpha x^2 + \beta x^3 \mid \alpha, \beta \in \mathbb{R} }$

    A2: $\mathrm{Span}(M) = { 0 }$

    A3: $\mathrm{Span}(M) = { x^2 , x^3 }$

    A4: $\mathrm{Span}(M) = { x^2 + x^3 }$

    Q3: Let $V = \mathcal{P}(\mathbb{R})$ be the real vector space consisting of polynomials. What is the dimension of the subspace $ { \alpha x + \beta x^2 + \gamma x \mid \alpha, \beta, \gamma }$?

    A1: 2

    A2: 3

    A3: 1

    A4: 0

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