Exercise
https://texercises.com/exercise/isomorphisms-and-dimensions/
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:
Two finite dimensional vector spaces V and W are isomorphic iff textdimVtextdimW.

Solution:
Proof of Longrightarrow. Suppose T:Vlongrightarrow W is an isomorphism. By a previous proposition textKerT and textImTW. By the rank theorem we have textdimWtextdim ImTtextdimV-textdim KerT textdimV Proof of Longleftarrow. Ase textdimVtextdimW and denote this dim by n. Let v_...v_n be a basis for V and w_...w_n a basis for W. Define a linear map T:Vlongrightarrow W by Tv_iw_iquad forall leq i leq n. By a previous theorem such a linear map exists. By the rank theorem textdim KerTn-textdim ImTn-n Longrightarrow T is also injective Longrightarrow T is an isomorphism.
Meta Information
\(\LaTeX\)-Code
Exercise:
Two finite dimensional vector spaces V and W are isomorphic iff textdimVtextdimW.

Solution:
Proof of Longrightarrow. Suppose T:Vlongrightarrow W is an isomorphism. By a previous proposition textKerT and textImTW. By the rank theorem we have textdimWtextdim ImTtextdimV-textdim KerT textdimV Proof of Longleftarrow. Ase textdimVtextdimW and denote this dim by n. Let v_...v_n be a basis for V and w_...w_n a basis for W. Define a linear map T:Vlongrightarrow W by Tv_iw_iquad forall leq i leq n. By a previous theorem such a linear map exists. By the rank theorem textdim KerTn-textdim ImTn-n Longrightarrow T is also injective Longrightarrow T is an isomorphism.
Contained in these collections:

Attributes & Decorations
Tags
dimension, eth, hs22, isomorphism, lineare algebra, proof, vector space
Content image
Difficulty
(3, default)
Points
0 (default)
Language
ENG (English)
Type
Proof
Creator rk
Decoration
File
Link