How do I test if a set of matrices is a subspace? I have some subsets of matrices defined for me, and I want to test if those are a subspace.
I know that the definition says that if:


*

*$x, y \in M \Rightarrow x+y \in M$

*$x \in M, \lambda \in \mathbb{R} \Rightarrow \lambda x \in M$


But how do I apply those rules on a subset of matrice?
I have for instance the first subset for which I should determine if it is a subspace or not:
$U_1 = \{ A \in \mathbb{R}^{2 \times 2}| \text{$A$ is a symmetric matrix}\}$
How do I do it?
 A: Firstly, there is no difference between the definition of a subspace of matrices or of one-dimensional vectors (i.e. scalars). Actually, a scalar can be considered as a matrix of dimension $1 \times 1$.
So as stated in your question, in order to show that set of points is a subspace of a bigger space M, one has to verify that :


*

*$x, y \in M \Rightarrow x+y \in M$

*$x \in M, \lambda \in \mathbb{R} \Rightarrow \lambda x \in M$


$x$ can be a scalar ( $x \in \mathbb{R}$), a matrix, a polynomial, a function...
In your case ( I will consider a generalization) :
$U_n = \{ A \in \mathbb{R}^{n \times n}| \text{$A$ is a symmetric matrix}\}$
To verify that  symmetric matrices form a subspace of $\mathbb{R}^{n \times n}$, firstly we consider $M,N \in U_n$ , as $M$ and $N$ are symmetric, by definition we have $M=M^T$ and $N = N^T$ ( $A^T$ is the transpose of $A$ )


*

*as $(M + N)^T = M^T + N^T = M+N$ , so $M + N$ is symmetric, in other words $M+N \in U_n,$

*and for $\lambda \in \mathbb{R}$, we have $(\lambda M)^T =\lambda M^T = \lambda M.$
As you can see, we can verify the two points easily in a general case.
In your case, because your matrices have a small dimension you can try to verify the two points element-wise (without using transpose properties ) by considering :
$$M := \begin{pmatrix}
a & c \\
c& b \end{pmatrix}$$ 
and
$$N:= \begin{pmatrix}
\alpha & \gamma \\
\gamma& \beta \end{pmatrix}$$ 
A: Yes. This is exactly a subspace. Since for any $A, B\in U_1$, A and B are symmetric matrixes, then $A+B$ is symmetric. So $A+B\in U_1$. For any $\lambda\in \mathbb{R}$, then $\lambda A$ is symmetric and $\lambda A\in U_1$.
 Hence by definition of a subspace, $U_1$ is a subspace.
A: A matrix is symmetric (i.e., is in $U_1$) iff $A^T = A$, or equivalently if it is in the kernel of the linear map $$M^{2 \times 2} \to M^{2 \times 2}, \qquad A \mapsto A^T - A,$$ but the kernel of any linear map is a subspace of the domain.
