• Title: Conservation of Dimension

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 28 | Conservation of Dimension

  • Bright video: https://youtu.be/XtiteY3n0Oc

  • Dark video: https://youtu.be/Ba4ycLfI5Rc

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: la28_sub_eng.srt missing

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  • Quiz Content

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

    A1: $f(\lambda x) = \lambda + f(x)$

    A2: $f(x+y) = f(x) + f(y)$

    A3: $f^{-1}$ exists.

    A4: $f^{-1}$ is linear.

    A5: $f^{-1}$ is bijective.

    Q2: If there is a bijective linear map $f: U \rightarrow V$ between two subspaces $U, V \subseteq \mathbb{R}^n$, then $\mathrm{dim}(U) = \mathrm{dim}(V)$. Is this correct?

    A1: Yes!

    A2: No, there are counterexamples.

    A3: One needs more information.

    Q3: Let $U = { \mathbf{x} \in \mathbb{R}^2 \mid \langle \mathbf{x} , \binom{2}{1} \rangle = 0}$. Now, take another subspace $V \subseteq U$ with $\mathrm{dim}(V) = 1$. Which claim is correct?

    A1: $V = U$.

    A2: $\mathrm{dim}(U) = 2$.

    A3: $\mathrm{dim}(U) = 3$.

    A4: $V$ is not a subspace.

    A5: $V = { 0 }$.

  • Last update: 2024-10

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