• Title: Column Picture of the Matrix-Vector Product

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 14 | Column Picture of the Matrix-Vector Product

  • Bright video: https://youtu.be/PZyTBvc9qxQ

  • Dark video: https://youtu.be/P56YTeY2YzM

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

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  • Subtitle on GitHub: la14_sub_eng.srt missing

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

    Q1: Which of the following claims is correct?

    A1: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is a linear combination of the columns of $A$.

    A2: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^m $, the vector $A \mathbf{x}$ is well-defined.

    A3: For $ A \in \mathbb{R}^{n \times m} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is well-defined.

    A4: For $ A \in \mathbb{R}^{m \times n} $ and $ \mathbf{x} \in \mathbb{R}^n $, the vector $A \mathbf{x}$ is a linear combination of the rows of $A$.

    Q2: Which of the following equations is correct?

    A1: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = x_1 \begin{pmatrix} 2 \ 0 \end{pmatrix} + x_2 \begin{pmatrix} 1 \ 1 \end{pmatrix} $$

    A2: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = x_1 \begin{pmatrix} 2 \ 1 \end{pmatrix} + x_2 \begin{pmatrix} 1 \ 0 \end{pmatrix} $$

    A3: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = 0 $$

    A4: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = \begin{pmatrix}x_1 \ x_2 \end{pmatrix} $$

    A5: $$ \begin{pmatrix} 2 & 1 \ 0 & 1 \end{pmatrix} \begin{pmatrix}x_1 \ x_2 \end{pmatrix} = \begin{pmatrix} 2 x_1 \ x_2 \end{pmatrix} $$

    Q3: A matrix $A \in \mathbb{R}^{m \times n}$ defines a map $f_A$. What is the correct definition?

    A1: $ f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m , , ~ \mathbf{x} \mapsto A \mathbf{x}$

    A2: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A \mathbf{x}$

    A3: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A - \mathbf{x}$

    A4: $ f_A: \mathbb{R}^m \rightarrow \mathbb{R}^n , , ~ \mathbf{x} \mapsto A + \mathbf{x}$

    A5: $ f_A: \mathbb{R}^n \rightarrow \mathbb{R}^m , , ~ \mathbf{x} \mapsto \mathbf{x}$

  • Last update: 2024-10

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