Exercise 285Linear algebra › Vector spaces and subspacesHigh schoolAdded 3 Nov 2026

Checking the vector space axioms

#vector space axioms#examples#counterexamples

Statement

Determine whether each of the following is a real vector space under the usual operations:

(a) $\mathbb{R}^2$ with addition $(x_1,y_1)+(x_2,y_2)=(x_1+x_2,y_1+y_2)$ and scalar multiplication $\lambda(x,y)=(\lambda x,\lambda y)$ ;
(b) $\mathbb{R}^2$ with the same addition but scalar multiplication $\lambda(x,y)=(0,0)$ ;
(c) The set of pairs $(x,y)$ with $x\ge0$ , under the usual operations.

Hint

Check each of the eight axioms. For (b), look at $1\cdot(x,y)$ . For (c), look at $(-1)\cdot(x,y)$ when $x>0$ .

Show solutionHide solution

Solution

(a) This is the standard $\mathbb{R}^2$ : all eight axioms hold. It is a vector space.

(b) Not a vector space. Indeed, for $(x,y)\neq(0,0)$ , $1\cdot(x,y)=(0,0)\neq(x,y)$ , violating the axiom $1\cdot v=v$ .

(c) Not a vector space. Take $(x,y)=(1,0)$ . Then $(-1)\cdot(1,0)=(-1,0)$ , which is not in the set because its first coordinate is negative. Scalar multiplication is not closed.

Conclusion. Only (a) is a real vector space.

Checking the vector space axioms

Statement

Solution

More problems in Vector spaces and subspaces