We denote the set of all \(2 \times 2\) matrices with real entries as \(M_{2\times 2}(\mathbb{R})\) or simply \(V\).
Let \(A, B, C \in V\) and \(c, d \in \mathbb{R}\) be arbitrary scalars.
1. Closure under Addition ✓ Holds
If \(A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\) and \(B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}\), then:
\[
A + B = \begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{pmatrix}
\]
Since the sum of real numbers is a real number, every entry in the resulting matrix is real. Thus, \(A+B \in V\).
2. Commutativity of Addition ✓ Holds
\[
A + B = \begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{pmatrix} = \begin{pmatrix} b_{11}+a_{11} & b_{12}+a_{12} \\ b_{21}+a_{21} & b_{22}+a_{22} \end{pmatrix} = B + A
\]
This relies on the commutativity of real number addition.
3. Associativity of Addition ✓ Holds
For matrices \(A, B, C\):
\[
(A + B) + C = \begin{pmatrix} (a_{11}+b_{11})+c_{11} & (a_{12}+b_{12})+c_{12} \\ (a_{21}+b_{21})+c_{21} & (a_{22}+b_{22})+c_{22} \end{pmatrix}
\]
\[
A + (B + C) = \begin{pmatrix} a_{11}+(b_{11}+c_{11}) & a_{12}+(b_{12}+c_{12}) \\ a_{21}+(b_{21}+c_{21}) & a_{22}+(b_{22}+c_{22}) \end{pmatrix}
\]
These are equal by the associativity of real number addition. Hence, \((A+B)+C = A+(B+C)\).
4. Additive Identity (Zero Vector) ✓ Holds
The zero matrix \( \mathbf{0} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \) belongs to \(V\) because all entries are real.
\[
A + \mathbf{0} = \begin{pmatrix} a_{11}+0 & a_{12}+0 \\ a_{21}+0 & a_{22}+0 \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = A
\]
Thus, \( \mathbf{0} \) acts as the additive identity.
5. Existence of Additive Inverse ✓ Holds
For any matrix \(A \in V\), define \(-A = \begin{pmatrix} -a_{11} & -a_{12} \\ -a_{21} & -a_{22} \end{pmatrix}\).
Since the negative of a real number is real, \(-A \in V\).
\[
A + (-A) = \begin{pmatrix} a_{11}+(-a_{11}) & a_{12}+(-a_{12}) \\ a_{21}+(-a_{21}) & a_{22}+(-a_{22}) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \mathbf{0}
\]
6. Closure under Scalar Multiplication ✓ Holds
For any scalar \(c \in \mathbb{R}\):
\[
cA = \begin{pmatrix} c \cdot a_{11} & c \cdot a_{12} \\ c \cdot a_{21} & c \cdot a_{22} \end{pmatrix}
\]
The product of two real numbers is real, so \(cA \in V\).
7. Distributivity of Scalar Multiplication over Vector Addition ✓ Holds
\begin{align*}
c(A + B) &= c \begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{pmatrix} \\
&= \begin{pmatrix} c(a_{11}+b_{11}) & c(a_{12}+b_{12}) \\ c(a_{21}+b_{21}) & c(a_{22}+b_{22}) \end{pmatrix} \\
&= \begin{pmatrix} c a_{11} + c b_{11} & c a_{12} + c b_{12} \\ c a_{21} + c b_{21} & c a_{22} + c b_{22} \end{pmatrix} \\
&= \begin{pmatrix} c a_{11} & c a_{12} \\ c a_{21} & c a_{22} \end{pmatrix} + \begin{pmatrix} c b_{11} & c b_{12} \\ c b_{21} & c b_{22} \end{pmatrix} \\
&= cA + cB
\end{align*}
8. Distributivity of Scalar Multiplication over Scalar Addition ✓ Holds
\begin{align*}
(c + d)A &= \begin{pmatrix} (c+d)a_{11} & (c+d)a_{12} \\ (c+d)a_{21} & (c+d)a_{22} \end{pmatrix} \\
&= \begin{pmatrix} c a_{11} + d a_{11} & c a_{12} + d a_{12} \\ c a_{21} + d a_{21} & c a_{22} + d a_{22} \end{pmatrix} \\
&= \begin{pmatrix} c a_{11} & c a_{12} \\ c a_{21} & c a_{22} \end{pmatrix} + \begin{pmatrix} d a_{11} & d a_{12} \\ d a_{21} & d a_{22} \end{pmatrix} \\
&= cA + dA
\end{align*}
9. Compatibility of Scalar Multiplication with Field Multiplication ✓ Holds
\begin{align*}
c(dA) &= c \begin{pmatrix} d a_{11} & d a_{12} \\ d a_{21} & d a_{22} \end{pmatrix} \\
&= \begin{pmatrix} c(d a_{11}) & c(d a_{12}) \\ c(d a_{21}) & c(d a_{22}) \end{pmatrix} \\
&= \begin{pmatrix} (cd) a_{11} & (cd) a_{12} \\ (cd) a_{21} & (cd) a_{22} \end{pmatrix} \quad \text{(by associativity of real multiplication)} \\
&= (cd)A
\end{align*}
10. Identity Element of Scalar Multiplication ✓ Holds
\[
1 \cdot A = \begin{pmatrix} 1 \cdot a_{11} & 1 \cdot a_{12} \\ 1 \cdot a_{21} & 1 \cdot a_{22} \end{pmatrix} = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = A
\]
Final Conclusion
Since all ten axioms for a vector space over \(\mathbb{R}\) are satisfied, the set \(V = M_{2\times 2}(\mathbb{R})\) with standard matrix addition and scalar multiplication is a vector space.
\(\boxed{\text{V is a vector space.}}\)
Note: This proof generalizes easily. The set of all \(m \times n\) matrices over \(\mathbb{R}\) forms a vector space denoted \(\mathbb{R}^{m \times n}\).
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