Exercise
https://texercises.com/exercise/examples-of-linear-subspaces/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
abcliste abc Let V be a vector space over K. Then _Vsubseteq V is a linear subspace. abc Fix bin K. Consider the subset U_b : x_x_x_in K^|x_+x_+x_b subseteq K^ bf claim: U_b subseteq K^ is a linear subspace iff b. abcliste

Solution:
abcliste abc by definition. abc Proof of Longrightarrow. Suppose U_b is a subspace of K^. If x_x_x_y_y_y_ in U_b then x_+x_+x_b y_+y_+y_b. Since U_b subseteq K^ is a subspace we have x_+y_ x_+y_ x_+y_b. But from above we also get x_+y_+x_+y_+x_+y_b+b Longrightarrow b+bb Longrightarrow b. Proof of Longleftarrow contradiction. Suppose b neq and x_x_x_y_y_y_in U_b. Then z_z_z_x_x_x_+y_y_y_ nicht in U_b da z_-z_+z_b+b neq b which implies that b must equal . abcliste
Report An Error
You are on texercises.com.
reCaptcha will only work on our main-domain \(\TeX\)ercises.com!
Meta Information
\(\LaTeX\)-Code
Exercise:
abcliste abc Let V be a vector space over K. Then _Vsubseteq V is a linear subspace. abc Fix bin K. Consider the subset U_b : x_x_x_in K^|x_+x_+x_b subseteq K^ bf claim: U_b subseteq K^ is a linear subspace iff b. abcliste

Solution:
abcliste abc by definition. abc Proof of Longrightarrow. Suppose U_b is a subspace of K^. If x_x_x_y_y_y_ in U_b then x_+x_+x_b y_+y_+y_b. Since U_b subseteq K^ is a subspace we have x_+y_ x_+y_ x_+y_b. But from above we also get x_+y_+x_+y_+x_+y_b+b Longrightarrow b+bb Longrightarrow b. Proof of Longleftarrow contradiction. Suppose b neq and x_x_x_y_y_y_in U_b. Then z_z_z_x_x_x_+y_y_y_ nicht in U_b da z_-z_+z_b+b neq b which implies that b must equal . abcliste
Contained in these collections

Similar exercises (48)
Title Creator Matched on
Addition of linear subspaces rk tags
Linear subspaces and complements rk tags
Span additional rk tags
Russel paradox rk tags
Algorithm to find a basis rk tags
more (43 more)
Attributes & Decorations
Tags
eth, hs22, linear subspace, lineare algebra, proof
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Decoration
Content image