Question

Let V=R2 . Define addition + on V by ( x1 , y1 ) + ( x2 , y2 ) = ( x1+x2 , y1+y2 ) and scalar multiplication . by r . (a , b) = (ra , 0). Check whether V satisfies all the
conditions for it to be a vector space over R with respect to these operations.

Answer 1

Answer on Question #68955 – Math – Linear Algebra

Question

Let $V = \mathbb{R}^2$. Define addition + on $V$ by

$$(x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$$

and scalar multiplication $\cdot$ by

$$r \cdot (a, b) = (ra, 0).$$

Check whether $V$ satisfies all the conditions for it to be a vector space over $\mathbb{R}$ with respect to these operations.

Solution

$V$ is not a vector space over $\mathbb{R}$ with respect to these operations. The axiom of identity element of scalar multiplication is not fulfilled.

In order to be a vector space, $V$ has to fulfill the axiom:

$$1 \cdot (a, b) = (a, b)$$

for arbitrary $(a, b) \in V$, where 1 denotes the multiplicative identity in $\mathbb{R}$.

But for example, for $(a, b) \in V$ with $b \neq 0$ for arbitrary $r \in \mathbb{R}$ we have

$$r \cdot (a, b) = (ra, 0) \neq (a, b).$$

Therefore, $V$ is not a vector space over $\mathbb{R}$ with respect to these operations.

**Answer**: $V$ is not a vector space over $\mathbb{R}$ with respect to these operations.

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