1

Let $B$ be a fixed matrix in $M_{n}(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. Define $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ by $$ T(A)=A...

Question

Let $B$ be a fixed matrix in $M_{n}(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. Define $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ by $$ T(A)=A B-B A . $$ Prove that $T$ is a linear operator on $M_{n}(\mathbb{F}) .$ Describe the kernel of $T$ if $B$ is scalar matrix. Find the range of $T$ in case $B$ is a diagonal matrix.

Let $B$ be a fixed matrix in $M_{n}(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. Define $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ by $$ T(A)=A B-B A . $$ Prove that $T$ is a linear operator on $M_{n}(\mathbb{F}) .$ Describe the kernel of $T$ if $B$ is scalar matrix. Find the range of $T$ in case $B$ is a diagonal matrix.



Answers

Let $B$ be a fixed matrix in $M_{n}(\mathbb{F})$, where $\mathbb{F}$ is an arbitrary field. Define $T: M_{n}(\mathbb{F}) \rightarrow M_{n}(\mathbb{F})$ by $$ T(A)=A B-B A . $$ Prove that $T$ is a linear operator on $M_{n}(\mathbb{F}) .$ Describe the kernel of $T$ if $B$ is scalar matrix. Find the range of $T$ in case $B$ is a diagonal matrix.


Similar Solved Questions

5 answers
X _ 64 Find all the vertical asymptotes (if any) of the graph of the function f (x) = x-4 A_ x} _ 64 The graph of the function f (x) has no vertical asymptotes_ X-4 B. x =- C. *=-4,4 D. x =-64 E. x=4
x _ 64 Find all the vertical asymptotes (if any) of the graph of the function f (x) = x-4 A_ x} _ 64 The graph of the function f (x) has no vertical asymptotes_ X-4 B. x =- C. *=-4,4 D. x =-64 E. x=4...
5 answers
The stock prices of Lwo COI[Ilies at (he ec of aY given yCAI TL( mOdeled wibh TandOI Variables. X ad that follow distribbion with joint dcnsiby [HcLion:
The stock prices of Lwo COI[Ilies at (he ec of aY given yCAI TL( mOdeled wibh TandOI Variables. X ad that follow distribbion with joint dcnsiby [HcLion:...
5 answers
7.71 7U2 I:04 EXc 1:04 .r.inirmobilyhilgll JSOivevde +
7.71 7 U2 I:04 EXc 1:04 .r.inir mobily hilgll JS Oivevde +...
5 answers
NAME CALCULAIION ASSIGNMENI MQLAR_MASS Calculate the molar mass of_each substance listed below: 1) CQ2 2) CzHso 3) Y2" (MoOj3 Be3 Nz NazS04 NaHcog 7) (NH_ '42C4H49 8) WA (Fe(CN}l3Hzo 10) C3ts0
NAME CALCULAIION ASSIGNMENI MQLAR_MASS Calculate the molar mass of_each substance listed below: 1) CQ2 2) CzHso 3) Y2" (MoOj3 Be3 Nz NazS04 NaHcog 7) (NH_ '42C4H49 8) WA (Fe(CN}l3 Hzo 10) C3ts0...
5 answers
IRINMR ` Unknown Spectra #@ Molr = [ASt 174.19 %( 562 "H 8.10LCioyThe lower spectrum in the 11O0-1500 cm"' region simply stronger fingerprint region _highlight
IRINMR ` Unknown Spectra #@ Molr = [ASt 174.19 %( 562 "H 8.10 LCioy The lower spectrum in the 11O0-1500 cm"' region simply stronger fingerprint region _ highlight...
5 answers
DeaLTanbttun0.085 mignabc leld:magnctic through thne cl Unitsrectanguler koop wilh sdes cm a1 ,pliced walholanoJCr{005
deaL Tanbttun 0.085 mignabc leld: magnctic through thne cl Units rectanguler koop wilh sdes cm a1 , pliced walh olano JCr{005...
5 answers
Suppose that X has the uniform distribution on the interval [0. 5] and that the random variable Y is defined by Y =0 il X<LY= if X>3 and Y =X otherwise: Sketch the cdf o Y.
Suppose that X has the uniform distribution on the interval [0. 5] and that the random variable Y is defined by Y =0 il X<LY= if X>3 and Y =X otherwise: Sketch the cdf o Y....
5 answers
Wukho thetelowing#tn 0l Une Gen cheieel Chootd= that acol IAGaonibien- excmoltv Ihakkhiznehren} & not ingenetic trke (txy are Kentical but hae d m. edantatsre onicobreni not Inllucncedby Iia hcasu Lult- enmplty that ichizoontyae at pcaee putly on ronn) edeo halet Kecnt Jhsuie unonainc CrrKnshtcrs mevhje
Wukho thetelowing#tn 0l Une Gen cheieel Chootd= that acol IAGaonibien- excmoltv Ihakkhiznehren} & not ingenetic trke (txy are Kentical but hae d m. edantatsre onicobreni not Inllucncedby Iia hcasu Lult- enmplty that ichizoontyae at pcaee putly on ronn) edeo halet Kecnt Jhsuie unonainc CrrKnshtcr...
5 answers
Are ONLY found in cardiac muscle; Intercalated discsUninucleiMultiple nucleiStriations
are ONLY found in cardiac muscle; Intercalated discs Uninuclei Multiple nuclei Striations...
5 answers
0A 1 7.3.25 Oezm %h4 2 Dt 34 cli Compldlc puli Homework: GuisuenheCuo) 20221Sp19 L 1 3 7.3 and Ha Choose tne CdEc[ anstel Delo; 3 1 1 3 8 8 1 =005, #ulgr Ian 1 Vv-eIlcen Cobos dovounao cnouan #hcULo WncOn lCcouit MH 1 Score: Question Help 1 1 1
0A 1 7.3.25 Oezm %h4 2 Dt 34 cli Compldlc puli Homework: GuisuenheCuo) 20221Sp19 L 1 3 7.3 and Ha Choose tne CdEc[ anstel Delo; 3 1 1 3 8 8 1 =005, #ulgr Ian 1 Vv-eIlcen Cobos dovounao cnouan #hcULo WncOn lCcouit MH 1 Score: Question Help 1 1 1...
5 answers
You are using a bicycle pump to fill a bicycle tire with air. It gets harder and harder to push the plunger on the pump the more air is in the tire. Explain what is going on.
You are using a bicycle pump to fill a bicycle tire with air. It gets harder and harder to push the plunger on the pump the more air is in the tire. Explain what is going on....
5 answers
Solve the following problem: 𝑀𝑎𝑥 𝑓 (𝑥, 𝑦) = (5 - 𝑥) 𝑥 + (10 - 𝑦)𝑦 - (𝑥 + 𝑦) 2 𝑆. 𝑎. 𝑦 ≤ 2, 𝑥 + 𝑦 ≥ 7/2
Solve the following problem: 𝑀𝑎𝑥 𝑓 (𝑥, 𝑦) = (5 - 𝑥) 𝑥 + (10 - 𝑦) 𝑦 - (𝑥 + 𝑦) 2 𝑆. 𝑎. 𝑦 ≤ 2, 𝑥 + 𝑦 ≥ 7/2...
1 answers
Lonsi der the ceal Valuel funchun ul}.y) witb Und ce vucia Les FF0c Cah deFini tiun 0 k Ulky) below ,. 6ins feal whe ther there @e exists a ceal-Valuel functiun U,y) Sud #hatt is funoKun aulytic in Somc DCc .If f(z)= ulx.y) +i vlx;y ) une Sucl' vU,y) an deteinc Scuh exists fine 0 VCX,y) If Sch VC,Y) does domain analy tu'ty P foc fcz > . the of nut eXist prove that 'k des @ not exist (i) ulx,y)= Xve_xy (6i5 Uly,Y)= Y} 3x-y (iii) ulx,Y)= Ac) (xtiy)
Lonsi der the ceal Valuel funchun ul}.y) witb Und ce vucia Les FF0c Cah deFini tiun 0 k Ulky) below ,. 6ins feal whe ther there @e exists a ceal-Valuel functiun U,y) Sud #hatt is funoKun aulytic in Somc DCc .If f(z)= ulx.y) +i vlx;y ) une Sucl' vU,y) an deteinc Scuh exists fine 0 VCX,y) If Sch ...

-- 0.022804--