Thread: Proof relating to Vector Spaces

1. Proof relating to Vector Spaces

Let $\displaystyle V$ be the set of all pairs $\displaystyle (x,y)$ or real numbers and $\displaystyle \Re$ be the feild of real numbers. Define $\displaystyle (x,y) + (x_{1},y_{1}) = (x+x_{1},y+y_{1}), c \cdot (x,y) = (cx,y)$. Prove that $\displaystyle V(\Re)$ is NOT a vector Space.

I can't seem to figure out which of the 10 conditions of Vector Spaces is being violated. My guess is that they involve the ones relating to the external composition since $\displaystyle V$ clearly is an abelian group, but I cannot disprove any of the conditions. Please help!

2. Re: Proof relating to Vector Spaces

The definition of addition is standard, so the problem is not there. There are only four axioms related to scalar multiplication (i.e., multiplication by a scalar). Double-check them carefully by writing everything explicitly. Finding the failing axiom is not hard.

3. Re: Proof relating to Vector Spaces

Originally Posted by sashikanth
Let $\displaystyle V$ be the set of all pairs $\displaystyle (x,y)$ or real numbers and $\displaystyle \Re$ be the feild of real numbers. Define $\displaystyle (x,y) + (x_{1},y_{1}) = (x+x_{1},y+y_{1}), c \cdot (x,y) = (cx,y)$. Prove that $\displaystyle V(\Re)$ is NOT a vector Space.
The linear algebra textbook by Larry Smith here Proposition 2.1: $\displaystyle 0\mathcal{A}=0.$ for any vector $\displaystyle \mathcal{A}$.
In this case what is $\displaystyle 0(1,1)=~?$.