Rank and basis
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!
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 finite dimensional VS over K of dimension ntextdimV mtextdimW. Let T:Vlongrightarrow W be a linear map with textrankTr. Recall that textrankT:textdim ImT Then exists a basis mathcalBv_...v_n of V and a basis mathcalCw_...w_m of W s.t. T_mathcalC^mathcalB leftarray@c|c@ matrix I_r matrix & _rn-r hline _m-rr & matrix _m-rn-r matrix arrayright
Solution:
Proof. Put l:textdim KerT. By the rank theorem rn-l. Let v_..v_l be a basis of textKerT and ext it to a basis v_..v_lv_l+...v_n of V. It will be useful to work below with the Basis mathcalBv_l+..v_nv_...v_n. In the proof of the rank theorem we have seen that Tv_l+...Tv_n form a basis for textImT. Define w_iTv_l+ i...r i.e. w_Tv_l+...w_rTv_l+r and ext this list to a basis mathcalC:w_...w_rw_r+...w_m of W. It now follows from the definitions that the statement is correct.
Let VW be finite dimensional VS over K of dimension ntextdimV mtextdimW. Let T:Vlongrightarrow W be a linear map with textrankTr. Recall that textrankT:textdim ImT Then exists a basis mathcalBv_...v_n of V and a basis mathcalCw_...w_m of W s.t. T_mathcalC^mathcalB leftarray@c|c@ matrix I_r matrix & _rn-r hline _m-rr & matrix _m-rn-r matrix arrayright
Solution:
Proof. Put l:textdim KerT. By the rank theorem rn-l. Let v_..v_l be a basis of textKerT and ext it to a basis v_..v_lv_l+...v_n of V. It will be useful to work below with the Basis mathcalBv_l+..v_nv_...v_n. In the proof of the rank theorem we have seen that Tv_l+...Tv_n form a basis for textImT. Define w_iTv_l+ i...r i.e. w_Tv_l+...w_rTv_l+r and ext this list to a basis mathcalC:w_...w_rw_r+...w_m of W. It now follows from the definitions that the statement is correct.