Let $V$ be a finite dimensional vector space over $\mathbf{R}$, and $V^{*}$ be the dual space of $V$.

If $W$ is a subspace of $V$, we define the subspace $\alpha(W)$ of $V^{*}$ by

$$\alpha(W)=\left\{f \in V^{*}: f(w)=0 \text { for all } w \text { in } W\right\}$$

Prove that $\operatorname{dim}(\alpha(W))=\operatorname{dim}(V)-\operatorname{dim}(W)$. Deduce that, if $A=\left(a_{i j}\right)$ is any real $m \times n$-matrix of rank $r$, the equations

$$\sum_{j=1}^{n} a_{i j} x_{j}=0 \quad(i=1, \ldots, m)$$

have $n-r$ linearly independent solutions in $\mathbf{R}^{n}$.