Exercise
https://texercises.com/exercise/kernel-and-image/
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:
Let VW be vector spaces over K and T:Vlongrightarrow W a linear map. abcliste abc The set textKerT:vin V| Tv_Wsubseteq V is a linear subspace of V. It is called the kernel of T. abc textImT:Tv|vin Vsubseteq W is a linear subspace of W. It is called the image of T. abcliste

Solution:
Proof. abcliste abc Let abin K v_v_in textKerT. Longrightarrow Tav_+bv_aTv_+bTv_a +b Longrightarrow av_+bv_extKerT. Also _VextKerT because T_V_W. Longrightarrow textKerTsubseteq V is a subspace abc Let abin K w_ w_in textImT. Longrightarrow exists v_v_in V s.t. Tv_w_ Tv_w_. Now Tav_+bv_aTv_+bTv_aw_+bw_. Longrightarrow aw_+bw_extImT. Also _Win textImT because T_V_W. Longrightarrow textImT is a subspace of W. 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:
Let VW be vector spaces over K and T:Vlongrightarrow W a linear map. abcliste abc The set textKerT:vin V| Tv_Wsubseteq V is a linear subspace of V. It is called the kernel of T. abc textImT:Tv|vin Vsubseteq W is a linear subspace of W. It is called the image of T. abcliste

Solution:
Proof. abcliste abc Let abin K v_v_in textKerT. Longrightarrow Tav_+bv_aTv_+bTv_a +b Longrightarrow av_+bv_extKerT. Also _VextKerT because T_V_W. Longrightarrow textKerTsubseteq V is a subspace abc Let abin K w_ w_in textImT. Longrightarrow exists v_v_in V s.t. Tv_w_ Tv_w_. Now Tav_+bv_aTv_+bTv_aw_+bw_. Longrightarrow aw_+bw_extImT. Also _Win textImT because T_V_W. Longrightarrow textImT is a subspace of W. abcliste
Contained in these collections

Similar exercises (49)
Title Creator Matched on
Russel paradox rk tags
Linearity Fibonacci sequences rk tags
Group characteristics rk tags
Row equivalence and solutions rk tags
Algorithm to find a basis rk tags
more (44 more)
Attributes & Decorations
Tags
eth, hs22, image, lineare algebra, proof
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Decoration
Content image