• Title: Set of Solutions

  • Series: Linear Algebra

  • Chapter: Matrices and linear systems

  • YouTube-Title: Linear Algebra 38 | Set of Solutions

  • Bright video: https://youtu.be/K9AnooBsjz4

  • Dark video: https://youtu.be/orIOFtmgG0c

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  • Quiz: Test your knowledge

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  • Subtitle on GitHub: la38_sub_eng.srt missing

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  • Timestamps

    00:00 Intro

    00:30 Solutions for systems of linear equations

    01:11 Visualization of range and kernel

    02:46 Existence needs range of A

    03:01 Uniqueness needs kernel of A

    03:13 The set of solutions is an affine subspace

    04:18 Concrete description of the set of solutions

    05:45 Proof of S = v + Ker(A)

    08:52 Row operations don’t change the set of solutions

    09:51 Recipe for finding the set of solutions

    11:00 Credits

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

    Q1: Which of the following solution sets is never possible for a system of linear equations $A \mathbf{x} = \mathbf{b}$?

    A1: There are exactly two solutions.

    A2: There is no solution.

    A3: There are infinitely many solutions.

    A4: There is exactly one solution.

    A5: The solution set is a one-dimensional subspace.

    Q2: Let $\mathbf{v}_0$ be a solution of $A \mathbf{x} = \mathbf{b}$ and $\widetilde{\mathbf{x}}$ be an element of $\mathrm{Ker}(A)$. Which claim is always correct?

    A1: $\mathbf{v}_0 + \widetilde{\mathbf{x}}$ is also a solution of $A \mathbf{x} = \mathbf{b}$.

    A2: $\mathbf{v}_0 + \widetilde{\mathbf{x}}$ is also an element of the kernel of $A$.

    A3: $\mathbf{v}_0 + \widetilde{\mathbf{x}}$ lies in the range of $A$.

    A4: $\mathbf{v}_0$ also lies in the kernel of $A$.

    Q3: What is the solution set of the system $$ \left( \begin{array}{cccc|c}1 & 0 & 3 & -1 & 0 \ 0 & 1 & 1 & -1 & 0 \ 0 & 0 & 0 & 0 & 1 \ \end{array} \right) $$

    A1: $\mathrm{S} = \emptyset$

    A2: $\mathrm{S} = { 0 }$

    A3: $\mathrm{S} = \left{ \begin{pmatrix} 2 \ 2 \ 1 \end{pmatrix} \right}$

    A4: $\mathrm{S} = \left{ \begin{pmatrix} 2 \ 2 \ 1 \ 0 \end{pmatrix} \right}$

    A5: $\mathrm{S} = \mathbb{R}^4 $

  • Last update: 2024-10

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