• Title: Basis of a subspace

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 24 | Basis of a subspace

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

  • Dark video: https://youtu.be/6VZiuEyTQVY

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: la24_sub_eng.srt missing

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  • Quiz Content

    Q1: Consider the subspace $U \subseteq \mathbb{R}^3$ defined by $U = \mathrm{Span}( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix} )$. What is not a basis of $U$?

    A1: The family given by $ \left( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 3 \end{pmatrix} \right) $

    A2: The family given by $ \left( \begin{pmatrix} 4 \ 2 \ 6 \end{pmatrix} \right) $

    A3: The family given by $ \left( \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix} \right) $

    A4: The family given by $ \left( \frac{1}{5} \begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix} \right) $

    Q2: Consider the subspace $U \subseteq \mathbb{R}^3$ defined by $U = \mathrm{Span} \left( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \right)$. What is a basis of $U$?

    A1: The family given by $ \left( \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} -1 \ 1 \ 0 \end{pmatrix} \right) $

    A2: The family given by $\left( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \right)$

    A3: The family given by $ \left( \begin{pmatrix} 0 \ 1 \ 0 \end{pmatrix} \right) $

    A4: The family given by $ \left( \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix}, \begin{pmatrix} 0 \ 0 \ 1 \end{pmatrix} \right) $

    Q3: Let $(\mathbf{u}, \mathbf{v})$ be a family of vectors in $\mathbb{R}^3$ that are linearly independent. Which claim is correct?

    A1: They form a basis of a subspace in $\mathbb{R}^3$.

    A2: They form not a basis of a subspace in $\mathbb{R}^3$.

    A3: They form a basis of $\mathbb{R}^3$.

    A4: One needs more information.

  • Last update: 2024-10

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