-
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
-
Quiz: Test your knowledge
-
Dark-PDF: Download PDF version of the dark video
-
Print-PDF: Download printable PDF version
-
Thumbnail (bright): Download PNG
-
Thumbnail (dark): Download PNG
-
Subtitle on GitHub: la38_sub_eng.srt missing
-
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