Homework SoursePlease give me all process 1 2 3 Let 5 denote the set of all
Please give me all process (1) (2) (3).
Let 5 denote the set of all vectors (x, y, z) in R^3 whose components satisfy the condition given. Determine whether S is a subspace of R^3. If S is a subspace, compute dim S. x = y or x = z. x^2 - y^2 = 0. x + y + z = 0 and x - y - z = 0.Solution
Let v1= (x1 ,y1 ,z1 ) and v2 = (x2 , y2 , z2 ) be two arbitrary elements of SThen v1 + v2 = (x1 ,y1 ,z1 ) + (x2 , y2 , z2 ) = ( x1 + x2 , y1 + y2 , z1 + z2 ) . Let us now check the conditions 1,2 and 3 above to see whether v1 + v2 is in S.
1. We have x1 + x2 = ( y1 or z1)+ (y2 or z2) = y1 + y2 or z1 + y2 or y1 + z2 or z1 + z2 . Thus, if x1 + x2 = y1 + y2 or, z1 + z2 , then the 1st condition is satisfied, but if x1 + x2 = z1 + y2 or, y1 + z2 , then the 1st condition is not satisfied..
2 [( x1 + x2)2 - ( y1 + y2)2] =( x12+2x1x2+x22 -y12 -2y1 y2-y22) = [ ( x12 - y12) + (x22 -y22) + 2( x1 x2 -y1 y2 ) ] = 2( x1 x2 -y1 y2 ) which is not necessarily equal to 0.
3 (x1 + y1 + z1 ) + ( x2 +2 + z2 ) = ( x1 + x2 , y1 + y2 , z1 + z2 ) Now, x1 + x2 + y1 + y2 + z1 + z2 = ( x1 + y1 + z1 )+ (x2 + y2 + z) = 0 + 0 = 0 and similarly (x1 + x2) -( y1 + y2 ) - ( z1 + z2 ) = ( x1 - y1 - z1 ) + (x2 - y2 - z2 ) = 0 + 0 = 0.
It is now apparent that addition in S satisfies only the condition number 3and not the 1st and the 2nd conditions. Therefore, S is not closed under addition and hence S is not a subspace of R3.
