Please prove the following 3 properties: 1. T is linear iff T(ax+by)=aT(y)+bT(y)

Question

Please prove the following 3 properties: 1. T is linear iff T(ax+by)=aT(y)+bT(y) for all x,ythats an element of V and C is an element of F. 2. If T is linear, then T(x-y)=T(x)-T(y) for all x,ythats an element of V. 3. T is linear iff, forx1,x2,....,xn thats an element ofV and a1,a2,....an thats anelement of F, we have              t(aixi)=aiT(xi). Please prove the following 3 properties: 1. T is linear iff T(ax+by)=aT(y)+bT(y) for all x,ythats an element of V and C is an element of F. 2. If T is linear, then T(x-y)=T(x)-T(y) for all x,ythats an element of V. 3. T is linear iff, forx1,x2,....,xn thats an element ofV and a1,a2,....an thats anelement of F, we have              t(aixi)=aiT(xi).

Explanation / Answer

CRAMSTER POLICY IS 1Q PERPOST
ANSWER I AND II Q.
GIVING GUIDANCE FOR IIIQ.
TRY ON YOUR OWN ON THAT BASIS
IF STILL IN DIFFICULTY PLEASE POST SEPARATELY.
Question Details: Please prove the following 3 properties: 1. T is linear iff T(ax+by)=aT(X)+bT(y) for all x,ythats an element of V and
C is an element of F. 2. If T is linear, then T(x-y)=T(x)-T(y) for all x,ythats an element of V. 3. T is linear iff, forx1,x2,....,xn thats an element ofV and a1,a2,....an thats
an element of F, we have.............t(aixi)=aiT(xi).
AS PER THE CRITERIA FOR L.T. WE HAVE
1.T[X+Y]=T[X]+T[Y]..........................C1
2.T[KX]=K*T[X]............................C2
SO NOW TPT
1. T is linear iff T(ax+by)=aT(y)+bT(y) for all x,y thats anelement of V and
C is an element of F.
PROPOSITION..
T(ax+by)=aT(X)+bT(y) ....................................3
TPT......T IS LINEAR
PUT A=1 AND B=1 IN EQN.3...WE GET
T[X+Y]=T[X]+T[Y]..........HENCE C1 IS OK....
PUT B=0 IN EQN.3....WE GET
T[KX]=K*T[X].........................HENCE C2 IS OK....
SO T IS LINEAR
CONVERSE
GIVEN T IS LINEAR ......TPT.....T(ax+by)=aT(X)+bT(y)
LHS = T[AX+BY] = T[C+D] SAY WHERE C=AX...AND...D=BY...WE GET
LHS=T[AX+BY] = T[C+D]=T[C]+T[D]=T[AX]+T[BY] = AT(X)+BT[Y]=RHS...PROVED
[FROM EQNS.C1 AND C2]

2. If T is linear, then T(x-y)=T(x)-T(y) for all x,y thats anelement of V. SAME WAY AS ABOVE ....
T[X-Y] = T[X]+T[-1*Y]=TX+(-1)T[Y]=T[X]-T[Y]

3. T is linear iff, forx1,x2,....,xn thats an element ofV and a1,a2,....an thats an element of F, we have.............t(aixi)=aiT(xi).
PROPOSITION
t(aixi)=aiT(xi).........TPT...............TIS LINEAR....
THIS IS AN EXTENSION OF PROOF WE GAVE UNDER 1.....VIZ
T is linear iff T(ax+by)=aT(X)+bT(y) for all x,y thats anelement of V and
C is an element of F.
FOLLOW THE SAME PROCEDURE TAKING 2 AT A TIME AND EXTENDING ONE BYONE IN A STEP
THAT IS LET
A2X2+A3X3+............AIXI]=BI*YI
APPLY THE ABOVE
T[A1X1+BIYI']=A1*T[X1]+BI*T[YI'] = A1T[X1]+T[BIYI']=
...NOW APPLY THE FORMULA FOR T[BIYI' ]....ETC....
OR USE MATHEMATICAL INDUCTION
IF STILL IN DIFFICULTY PLEASE POST SEPARATELY...
SO NOW TPT
1. T is linear iff T(ax+by)=aT(y)+bT(y) for all x,y thats anelement of V and
C is an element of F.
PROPOSITION..
T(ax+by)=aT(X)+bT(y) ....................................3
TPT......T IS LINEAR
PUT A=1 AND B=1 IN EQN.3...WE GET
T[X+Y]=T[X]+T[Y]..........HENCE C1 IS OK....
PUT B=0 IN EQN.3....WE GET
T[KX]=K*T[X].........................HENCE C2 IS OK....
SO T IS LINEAR
CONVERSE
GIVEN T IS LINEAR ......TPT.....T(ax+by)=aT(X)+bT(y)
LHS = T[AX+BY] = T[C+D] SAY WHERE C=AX...AND...D=BY...WE GET
LHS=T[AX+BY] = T[C+D]=T[C]+T[D]=T[AX]+T[BY] = AT(X)+BT[Y]=RHS...PROVED
[FROM EQNS.C1 AND C2]

2. If T is linear, then T(x-y)=T(x)-T(y) for all x,y thats anelement of V. SAME WAY AS ABOVE ....
T[X-Y] = T[X]+T[-1*Y]=TX+(-1)T[Y]=T[X]-T[Y]

3. T is linear iff, forx1,x2,....,xn thats an element ofV and a1,a2,....an thats

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