Consider the set of all points X= (x1.x2) and Y= (yl. y2) in R2 plane with...

  

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Consider the set of all points X= (x1.x2) and Y= (yl. y2) in R2 plane with the operation of abstract addition in defined as XOY-(x1,x2) (yl, y2)=(x1+yl-1.x2 + y2-1). and the operation of abstract scalar multiplication is defined as coX(cxl-c+ 1.ex2 - + 1). where e is a scalar, and plus +" and the dot" "stand for standard operations of addition and multiplication of real numbers. These abstract operations obviously satisfy the first two axioms for vector spaces, namely, the closure of the set R² with respect to addition and scalar multiplication. Verify whether each of the following 3 axioms for vector spaces can be satisfied: (A). Existence of a vector 0 (zero) ER² such that for every XER², we shall have X-X Find the zero-vector 0 if it exists. (B) For every XER² there always exists its additive inverse vector IER² such that Xe 1-0 (where 0 is the zero vector). Find the additive inverse if it exists. (C). Distributivity of scalar multiplication over vector addition: For any vectors and YER2, and any scalar CER, c (X@r) (c-x) @ (e-Y): Consider the set of all points X= (x1.x2) and Y= (yl. y2) in R2 plane with the operation of abstract addition in defined as XOY-(x1,x2) (yl, y2)=(x1+yl-1.x2 + y2-1). and the operation of abstract scalar multiplication is defined as coX(cxl-c+ 1.ex2 - + 1). where e is a scalar, and plus +" and the dot" "stand for standard operations of addition and multiplication of real numbers. These abstract operations obviously satisfy the first two axioms for vector spaces, namely, the closure of the set R² with respect to addition and scalar multiplication. Verify whether each of the following 3 axioms for vector spaces can be satisfied: (A). Existence of a vector 0 (zero) ER² such that for every XER², we shall have X-X Find the zero-vector 0 if it exists. (B) For every XER² there always exists its additive inverse vector IER² such that Xe 1-0 (where 0 is the zero vector). Find the additive inverse if it exists. (C). Distributivity of scalar multiplication over vector addition: For any vectors and YER2, and any scalar CER, c (X@r) (c-x) @ (e-Y):