Show that the set of all vectors (x,y,z) such that x+y+z=0 is a subspace of R3.
Solution
Show that the set of all vectors (x,y,z) such that x+y+z=0 is a subspace of R3.
THE NECESSARY AND SUFFICIENT CONDITION FOR A SET S TO BE SUB SPACE OF V IS THAT ........
S3=P*S1+Q*S2 SHALL BE AN ELEMENT OF S ....WHERE .....
S1,S2, ARE ANY 2 ELEMENTS OF SET S AND P AND Q ARE ANY 2 SCALARS.....
LET S1=[X1,Y1,Z1]..........................S2=[X2,Y2,Z2] .......THEY ARE ELEMENTS OF S ...SO AS PER GIVEN CONDITION , WE HAVE ...
X1+Y1+Z1=0........................................1......................
X2+Y2+Z2=0........................................2...........................
S3=P*S1+Q*S2=P[X1,Y1,Z1]+Q[X2,Y2,Z2] = [PX1+QX2 , PY1+QY2 , PZ1+QZ2] ..
SO NOW LET US CHECK IF
PX1+QX2 + PY1+QY2 + PZ1+QZ2 = 0 OR NOT ....
PX1+QX2 + PY1+QY2 + PZ1+QZ2 =P[X1+Y1+Z1]+Q[X2+Y2+Z2]=P*0+Q*0=0
SO................. S3=P*S1+Q*S2 IS AN ELEMENT OF S ...........
SO S FORMS A SUB SPACE ..........PROVED ............................
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Show that the set of all vectors (x-y-z) such that x+y+z-0 is a subspa.docx
1. Show that the set of all vectors (x,y,z) such that x+y+z=0 is a subspace of R3.
Solution
Show that the set of all vectors (x,y,z) such that x+y+z=0 is a subspace of R3.
THE NECESSARY AND SUFFICIENT CONDITION FOR A SET S TO BE SUB SPACE OF V IS THAT ........
S3=P*S1+Q*S2 SHALL BE AN ELEMENT OF S ....WHERE .....
S1,S2, ARE ANY 2 ELEMENTS OF SET S AND P AND Q ARE ANY 2 SCALARS.....
LET S1=[X1,Y1,Z1]..........................S2=[X2,Y2,Z2] .......THEY ARE ELEMENTS OF S ...SO AS PER GIVEN
CONDITION , WE HAVE ...
X1+Y1+Z1=0........................................1......................
X2+Y2+Z2=0........................................2...........................
S3=P*S1+Q*S2=P[X1,Y1,Z1]+Q[X2,Y2,Z2] = [PX1+QX2 , PY1+QY2 , PZ1+QZ2] ..
SO NOW LET US CHECK IF
PX1+QX2 + PY1+QY2 + PZ1+QZ2 = 0 OR NOT ....
PX1+QX2 + PY1+QY2 + PZ1+QZ2 =P[X1+Y1+Z1]+Q[X2+Y2+Z2]=P*0+Q*0=0
SO................. S3=P*S1+Q*S2 IS AN ELEMENT OF S ...........
SO S FORMS A SUB SPACE ..........PROVED ............................
![Show that the set of all vectors (x,y,z) such that x+y+z=0 is a subspace of R3.
Solution
Show that the set of all vectors (x,y,z) such that x+y+z=0 is a subspace of R3.
THE NECESSARY AND SUFFICIENT CONDITION FOR A SET S TO BE SUB SPACE OF V IS THAT ........
S3=P*S1+Q*S2 SHALL BE AN ELEMENT OF S ....WHERE .....
S1,S2, ARE ANY 2 ELEMENTS OF SET S AND P AND Q ARE ANY 2 SCALARS.....
LET S1=[X1,Y1,Z1]..........................S2=[X2,Y2,Z2] .......THEY ARE ELEMENTS OF S ...SO AS PER GIVEN
CONDITION , WE HAVE ...
X1+Y1+Z1=0........................................1......................
X2+Y2+Z2=0........................................2...........................
S3=P*S1+Q*S2=P[X1,Y1,Z1]+Q[X2,Y2,Z2] = [PX1+QX2 , PY1+QY2 , PZ1+QZ2] ..
SO NOW LET US CHECK IF
PX1+QX2 + PY1+QY2 + PZ1+QZ2 = 0 OR NOT ....
PX1+QX2 + PY1+QY2 + PZ1+QZ2 =P[X1+Y1+Z1]+Q[X2+Y2+Z2]=P*0+Q*0=0
SO................. S3=P*S1+Q*S2 IS AN ELEMENT OF S ...........
SO S FORMS A SUB SPACE ..........PROVED ............................](https://image.slidesharecdn.com/showthatthesetofallvectorsx-y-zsuchthatxyz-0isasubspa-230209002405-6e0e2a8c/85/Show-that-the-set-of-all-vectors-x-y-z-such-that-x-y-z-0-is-a-subspa-docx-1-320.jpg)
