• Title: Example of Basis Isomorphism

  • Series: Abstract Linear Algebra

  • Chapter: General vector spaces

  • YouTube-Title: Abstract Linear Algebra 6 | Example of Basis Isomorphism

  • Bright video: https://youtu.be/xawP1WXokzM

  • Dark video: https://youtu.be/W2dXg_P8_gY

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

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  • Subtitle on GitHub: ala06_sub_eng.srt missing

  • Timestamps

    00:00 Introduction

    00:33 Span in the function space

    01:27 Is this a basis?

    02:03 Checking for linear independence

    05:40 Reformulating as matrix-vector multiplication

    06:52 Uniquely solvable?

    08:52 Basis isomorphism

    11:16 Credits

  • Subtitle in English (n/a)
  • Quiz Content

    Q1: Let $V$ be the real vector space of functions $f: \mathbb{R} \rightarrow \mathbb{R}$. Are $\cos$ and $\sin$ linearly independent?

    A1: Yes!

    A2: No!

    A3: Only for $x = 0$.

    Q2: Let $V$ be the real vector space of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $U$ the subspace spanned by the two vectors $\cos$ and $\sin$. For $\mathcal{B} = (\cos, \sin)$ let $\Phi_{\mathbb{B}}$ be the corresponding basis isomorphism. What is $\Phi_{\mathbb{B}}( 3 \sin + 7 \cos)$?

    A1: $\begin{pmatrix} 7 \ 3 \end{pmatrix}$

    A2: $\begin{pmatrix} 3 \ 7 \end{pmatrix}$

    A3: $\begin{pmatrix} 7 \ 3 \ 0 \end{pmatrix}$

    A4: $\begin{pmatrix} 0 \ 3 \ 7 \end{pmatrix}$

  • Last update: 2024-10

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