Let vector A=
⎛ a11 a12 ⎞
⎝ a21 a22 ⎠
and B=
⎛ b11 b12 ⎞
⎝ b21 b22 ⎠
where aij and bij are real expressions.
Addition: A+B=
⎛ a11+b11 a12+b12 ⎞
⎝ a21+b21 a22+b22 ⎠
B+A=
⎛ b11+a11 b12+a12 ⎞
⎝ b21+a21 b22+a22 ⎠
But since a11+b11=b11+a11 under the axioms for real numbers and expressions, A+B=B+A (commutativity).
Also since x+y ∈ ℝ for x,y ∈ ℝ and A and B ∈ V, it follows that, since the vectors can be expressed as matrices composed of real numbers or expressions, their sum also belongs to V, the vector space.
For multiplication AB consists of multiplication and addition (the sum of products). The product is:
⎛ a11b11+a12b21 a11b12+a12b22 ⎞
⎝ a21b11+a22b21 a21b12+a12b22 ⎠
Each sum of the products involves multiplication of real numbers, and we know that the axioms governing real numbers and expressions ensures that the general product of two quantities x and y, for both in ℝ, the product is also in ℝ. The sum of two such quantities is also in ℝ. Each element of the vector product is therefore in ℝ, implying that the vector product is in V, the vector space.