verify that (V, +, x) is a vector space.

Question

Prove that a 2x2 matrix under addition and multiplication is a vector space

Answer 1

Let vector A=

⎛ a₁₁ a₁₂ ⎞

⎝ a₂₁ a₂₂ ⎠

and B=

⎛ b₁₁ b₁₂ ⎞

⎝ b₂₁ b₂₂ ⎠

where a_ij and b_ij are real expressions.

Addition: A+B=

⎛ a₁₁+b₁₁ a₁₂+b₁₂ ⎞

⎝ a₂₁+b₂₁ a₂₂+b₂₂ ⎠

B+A=

⎛ b₁₁+a_{11 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.