Use the closure property a) if u and v are vectors in W then u+v is in W b) if k
ID: 2939973 • Letter: U
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Use the closure property a) if u and v are vectors in W then u+v is in W b) if k is any scalar and u is any vector in W, then ku is inW. I have to Vectors Vector 1 , all vectors of the form (a,b,c,) where b= a+c Vector 2 all vectors of the form (a,b,c) where b= a+c+1 Cramster says that vector one should be closed under addtionand should be a subspace but it also says that vector 2 is notclosed under addtion and is not a subspace. Can somebodyexplain the difference by actually doing the problems? a) if u and v are vectors in W then u+v is in W b) if k is any scalar and u is any vector in W, then ku is inW. I have to Vectors Vector 1 , all vectors of the form (a,b,c,) where b= a+c Vector 2 all vectors of the form (a,b,c) where b= a+c+1 Cramster says that vector one should be closed under addtionand should be a subspace but it also says that vector 2 is notclosed under addtion and is not a subspace. Can somebodyexplain the difference by actually doing the problems?Explanation / Answer
QuestionDetails: a) if u and v are vectors in W then u+v is in W b) if k is any scalar and u is any vector in W, then ku is inW. I have to Vectors Vector 1 , all vectors of the form (a,b,c,) where b= a+cTHAT IS THE GENERAL FORM OF VECTOR IS
V=[A,A+C,C]
CONSIDER 2 ELEMENTS IN V...V1 AND V2
V1=[A1,A1+C1,C1]....V2=[A2,A2+C2,C2]
TO CHECK IF XV1+YV2 LIES IN V..
IF IT IS AN ELEMENT OF V THEN IT IS A SUB SPACE ...OTHERWISENOT
....THAT IS THE TEST
XV1+YV2=X[A1,A1+C1,C1]+Y[A2,A2+C2,C2]=
=[XA1+YA2,XA1+XC1+YA2+YC2,XC1+YC2]
IF WE TAKE .....A3=XA1+YA2....C3=XC1+YC2, WE NOTE THAT THIS IS
OF THE FORM [A3,A3+C3,C3],...HENCE THIS IS AN ELEMENT OFV
HENCE THIS IS A SUBSPACE.
Vector 2 all vectors of the form (a,b,c) where b= a+c+1
THAT IS THE GENERAL FORM OF VECTOR IS
V=[A,A+C+1,C]
CONSIDER 2 ELEMENTS IN V...V1 AND V2
V1=[A1,A1+C1+1,C1]....V2=[A2,A2+C2+1,C2]
TO CHECK IF XV1+YV2 LIES IN V..
IF IT IS AN ELEMENT OF V THEN IT IS A SUB SPACE ...OTHERWISENOT
....THAT IS THE TEST
XV1+YV2=X[A1,A1+C1+1,C1]+Y[A2,A2+C2+1,C2]=
=[XA1+YA2,XA1+XC1+1+YA2+YC2+1,XC1+YC2]
IF WE TAKE .....A3=XA1+YA2....C3=XC1+YC2, WE NOTE THAT THIS IS
OF THE FORM [A3,A3+C3+2,C3],...
IT IS NOT OF THE TYPE [A3,A3+C3+1,C3]
HENCE THIS IS NOT AN ELEMENT OF V
HENCE THIS IS NOT A SUBSPACE.
Cramster says that vector one should be closed under addtionand should be a subspace but it also says that vector 2 is notclosed under addtion and is not a subspace. Can somebodyexplain the difference by actually doing the problems?
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