I have thoroughly confused myself and need lots of help. The question is: Determ
ID: 2938372 • Letter: I
Question
I have thoroughly confused myself and need lots of help. The question is: Determine whether the following sets form subspaces ofR^2. A) {(X1, X2)^T | X1+X2=0} B) {(X1, X2)^T | X1*X2=0} C) {(X1, X2)^T | X1=3X2} D) {(X1, X2)^T | |X1| = |X2|} E) {(X1, X2)^T | X21 =X22} The book says that A and C are subspaces; B, D, and E arenot. I cannot even figure out A. I know that X1=-X2 andtried to make S1 = (X1, -X1) and S2=(-X2, X2) but I get that it isnot a subspace. Can someone help? Will rate lifesaverasap...I am not stingy with ratings! I have thoroughly confused myself and need lots of help. The question is: Determine whether the following sets form subspaces ofR^2. A) {(X1, X2)^T | X1+X2=0} B) {(X1, X2)^T | X1*X2=0} C) {(X1, X2)^T | X1=3X2} D) {(X1, X2)^T | |X1| = |X2|} E) {(X1, X2)^T | X21 =X22} The book says that A and C are subspaces; B, D, and E arenot. I cannot even figure out A. I know that X1=-X2 andtried to make S1 = (X1, -X1) and S2=(-X2, X2) but I get that it isnot a subspace. Can someone help? Will rate lifesaverasap...I am not stingy with ratings!Explanation / Answer
QuestionDetails: I have thoroughly confused myself and need lots of help. The question is: Determine whether the following sets form subspaces ofR^2. A) {(X1, X2)^T | X1+X2=0} B) {(X1, X2)^T | X1*X2=0} C) {(X1, X2)^T | X1=3X2} D) {(X1, X2)^T | |X1| = |X2|} E) {(X1, X2)^T | X21 =X22} The book says that A and C are subspaces; B, D, and E arenot. I cannot even figure out A. I know that X1=-X2 andtried to make S1 = (X1, -X1) and S2=(-X2, X2) but I get that it isnot a subspace. Can someone help? Will rate lifesaverasap...I am not stingy with ratings!METHOD.....W.R.T. PROBLEM A.
1. TAKE ANY 2 VECTORS IN THE GIVEN SET S.....SAY S1 AND S2SATISFYING THE GIVEN CRITERIA.....
A) {(X1, X2)^T | X1+X2=0}
LET
S1=[X1,X2]..............X1+X2=0................1
S2=[Y1,Y2]..............Y1+Y2=0.................2
2. TAKE ANY 2 SCALARS P AND Q SAY
3.CHECK WHETHER S3= PS1+QS2 IS AN ELEMENT OF S
WE HAVE HERE
S3=PS1+QS2=P[X1,X2]+Q[Y1,Y2]=[PX1+QY1,PX2+QY2]=[R1,R2] SAY
WE KNOW THAT S3=[R1,R2] WILL BE AN ELEMENT OF S IFR1+R2=0
LET US CHECK
PX1+QY1-PX2-QY2=P[X1-X2]+Q[Y1-Y2]=P*0+Q*0=0 AS PER EQNS. 1 AND 2.
HENCE S3= PS1+QS2 IS AN ELEMENT OF S
SO S IS A SUBSPACE OF R..........GOT IT ?
================================
NOW LET US TRY IT IN
B) {(X1, X2)^T | X1*X2=0}
LET
S1=[X1,X2]..............X1*X2=0................3
S2=[Y1,Y2]..............Y1*Y2=0.................4
2. TAKE ANY 2 SCALARS P AND Q SAY
3.CHECK WHETHER S3= PS1+QS2 IS AN ELEMENT OF S
WE HAVE HERE
S3=PS1+QS2=P[X1,X2]+Q[Y1,Y2]=[PX1+QY1,PX2+QY2]=[R1,R2] SAY
WE KNOW THAT S3=[R1,R2] WILL BE AN ELEMENT OF S IFR1*R2=0
LET US CHECK
S3=(PX1+QY1)*(PX2+QY2)=P[X1X2]+PQ[X1Y2+X2Y1]+Q[Y1Y2]=
S3=P*0+PQ[X1Y2+X2Y1]+Q*0=PQ[X1Y2+X2Y1]......FROM EQNS.3 AND 4
THIS MAY NOT BE ZERO
HENCE S3= PS1+QS2 IS NOT AN ELEMENT OF S
SO S IS NOT A SUBSPACE OF R..........GOT IT ?
YOU CAN TRY OTHERS SIMILARLY.
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