Let $\mathbb{M}$ be the vector space of $m \times p$ real matrices. For a vector subspace $S\subseteq \mathbb{M}$, denote by $\delta(S)$ the dimension of the vector space generated by all columns of all matrices in $S$. 
Say that a vector subspace $T\subseteq \mathbb{M}$ is a $\emph{covering matrix space}$ if 
\[ \bigcup_{A\in T, A\ne \mathbf{0}} \ker A =\mathbb{R}^p \]
Such a $T$ is minimal if it doesn't contain a proper vector subspace $S\subset T$ such that $S$ is also a covering matrix space.
[list]
(a) (8 points) Let $T$ be a minimal covering matrix space and let $n=\dim (T)$
Prove that
\[ \delta(T)\le \dbinom{n}{2} \]
(b) (2 points) Prove that for every integer $n$ we can find $m$ and $p$, and a minimal covering matrix space $T$ as above such that $\dim T=n$ and $\delta(T)=\dbinom{n}{2}$[/list]
### Part (a)

1. **Define the problem and notation:**
   Let \( T \) be a minimal covering matrix space and let \( n = \dim(T) \). We need to prove that:
   \[
   \delta(T) \le \binom{n}{2}
   \]
   where \( \delta(T) \) is the dimension of the vector space generated by all columns of all matrices in \( T \).

2. **Understand the covering matrix space:**
   Since \( T \) is a covering matrix space, for any \( x \in \mathbb{R}^p \), there exists a non-zero matrix \( A \in T \) such that \( Ax = 0 \). This implies that the union of the kernels of all non-zero matrices in \( T \) covers \( \mathbb{R}^p \).

3. **Construct a basis:**
   We can construct a basis \( \{b_1, b_2, \ldots, b_p\} \) of \( \mathbb{R}^p \) and corresponding matrices \( A_1, A_2, \ldots, A_p \in T \) such that \( A_i \) are linearly independent and the only elements of \( T \) that zero \( b_i \) are scalar multiples of \( A_i \).

4. **Inductive construction:**
   Suppose we have pairs \( (A_1, b_1), (A_2, b_2), \ldots, (A_k, b_k) \). We choose a proper vector subspace \( S_k \subset T \) that covers \( \{b_1, b_2, \ldots, b_k\} \) and has dimension \( n-1 \). Since \( S_k \) does not cover the entire \( \mathbb{R}^p \), we can choose \( b_{k+1} \) that is zeroed by exactly one element \( A_{k+1} \notin S_k \). This ensures that \( A_{k+1} \) does not zero any linear combination in \( \{b_1, \ldots, b_k\} \), and \( A_1, \ldots, A_{k+1} \) are linearly independent.

5. **Dimension argument:**
   We show that:
   \[
   \dim((T[b_1] + T[b_2] + \ldots + T[b_k]) \cap T[b_{k+1}]) \ge k
   \]
   and:
   \[
   \dim(T[b_1] + T[b_2] + \ldots + T[b_{k+1}]) \le \dim(T[b_1] + \ldots + T[b_k]) + n - k - 1
   \]
   Since \( \dim(T[b_i]) \le n-1 \), the bound follows.

6. **Conclusion:**
   By the above construction and dimension argument, we conclude that:
   \[
   \delta(T) \le \binom{n}{2}
   \]

### Part (b)

1. **Construct an example:**
   We provide an example of a minimal covering matrix space with \( p = n \) and \( m = \binom{n}{2} \). Take the usual basis of \( \mathbb{R}^n \), \( e_1 = (1, 0, 0, \ldots, 0) \), \( e_2 = (0, 1, 0, \ldots, 0) \), \ldots, \( e_n = (0, 0, \ldots, 1) \).

2. **Define matrices:**
   We take \( n \) linearly independent linear transformations from \( \mathbb{R}^n \) to \( \mathbb{R}^{\frac{n(n-1)}{2}} \) with matrices \( A_1, A_2, \ldots, A_n \) which satisfy the following:
   - \( A_i e_i = 0 \)
   - \( A_i e_j + A_j e_i = 0 \)
   - For different unordered pairs \( \{i, j\} \), \( A_i e_j = -A_j e_i \) are \( \binom{n}{2} \) different basis elements of \( \mathbb{R}^{\frac{n(n-1)}{2}} \).

3. **Verify the construction:**
   For these matrices, we get that:
   \[
   (x_1 A_1 + x_2 A_2 + \ldots + x_n A_n)(x_1 e_1 + x_2 e_2 + \ldots + x_n e_n) = 0
   \]
   which shows that this construction works.

4. **Example for \( n = 3 \):**
   For \( n = 3 \), we have:
   \[
   T = \left\{
   \begin{pmatrix}
   0 & 1 & 0 \\
   0 & 0 & 1 \\
   0 & 0 & 0
   \end{pmatrix},
   \begin{pmatrix}
   -1 & 0 & 0 \\
   0 & 0 & 0 \\
   0 & 0 & 1
   \end{pmatrix},
   \begin{pmatrix}
   0 & -1 & 0 \\
   0 & 0 & -1 \\
   0 & 0 & 0
   \end{pmatrix}
   \right\}
   \]

5. **Conclusion:**
   This example shows that for every integer \( n \), we can find \( m \) and \( p \), and a minimal covering matrix space \( T \) such that \( \dim(T) = n \) and \( \delta(T) = \binom{n}{2} \).

The final answer is \( \boxed{ \delta(T) \le \binom{n}{2} } \) for part (a) and \(\delta(T) = \binom{n}{2}\) for part (b).