• Title: Matrix Representation for Linear Maps

  • Series: Abstract Linear Algebra

  • YouTube-Title: Abstract Linear Algebra 25 | Matrix Representation for Linear Maps

  • Bright video: https://youtu.be/2zrG8cvdt5E

  • Dark video: https://youtu.be/r9bV_9xOpyw

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ala25_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $V$ be a $3$-dimensional real vector space with basis $\mathcal{B}$ and $W$ be a $4$-dimensional real vector space with $\mathcal{C}$. What is correct for a linear map $\ell: V \rightarrow W$?

    A1: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(4\times 3)$-matrix with real entries.

    A2: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(3\times 4)$-matrix with real entries.

    A3: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(3\times 3)$-matrix with complex entries.

    A4: The matrix representation $\ell_{ \mathcal{C} \leftarrow \mathcal{B} }$ is a $(4\times 4)$-matrix with complex entries.

    Q2: Can the matrix $\begin{pmatrix} 1 & 0 & 3 \ 2 & 1 & 1 \end{pmatrix}$ be a matrix representation of the linear map $ \ell: \mathcal{P}_2(\mathbb{R}) \rightarrow \mathcal{P}_1(\mathbb{R}) $ given by $\ell(p) = p^\prime$.

    A1: Yes, if one chooses suitable bases $\mathcal{B}$ and $\mathcal{C}$.

    A2: No, the matrix is of wrong size.

    A3: No, because the matrix has an entry given by 3.

    A4: No, the matrix has only real entries.

  • Last update: 2024-10

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