Let \(m,n\) be positive integers and \(\mathbb{F}\) a commutative field. A linear map \(\phi : M_{m,n}(\mathbb{F}) \rightarrow M_{m,n}(\mathbb{F})\) which preserves certain ``something'' (a property, a subset, etc.) is called a linear preserver. A common question is to ask whether a linear preserver is of standard form, i.e., there exist invertible matrices \(S\) and \(T\) such that \(\phi(A)=TAS\) or \(\phi(A)=TA^tS\) (this could happen if \(m=n\)). A linear map \(h: M_{m,n}(\mathbb{F}) \rightarrow \mathbb{F}^m\) is also said to be of standard form if there exists an invertible \(T\) and a vector \(u\) such that \(h(A)=TAu\) or \(h(A)=TA^tu\). If \(\phi\) is a linear preserver of standard form, then the map \(\phi_x(A)=\phi(A)x\) is also standard for any \(x\in \mathbb{F}^m\). In this article, the authors prove that the converse is almost true: Theorem 3.2: If \(\phi_x(A)=\phi(A)x\) is also standard for any \(x\in \mathbb{F}^m\), then either \(\phi\) is standard or \(\phi(A)=\alpha(Au)\), where \(u\in \mathbb{F}^m\) and \(\alpha\) is a bijective linear map from \(\mathbb{F}^m\) onto an \(m\)-dimensional subspace of full rank matrices. In certain sense, we have that a local property implies a global property. The authors go on to prove some results on linear maps preserving full rank matrices (Theorem 4.1), preserving rank one matrices (Theorem 4.2) and preserving unitary matrices (Theorem 4.4). (It is worth to mention that: In order to prove Theorem 3.2, the authors state a nice result in Theorem 2.1, which says that a linear map \(h: M_{m,n}(\mathbb{F}) \rightarrow \mathbb{F}^m\) is of the form \[ h(B)=T_1Bx_1+\cdots+T_rBx_r \] or, in the case \(m=n\), \[ h(B)=T_1B^tx_1+\cdots+T_rB^tx_r, \] where \(\{T_1,\ldots,T_r\}\) and \(\{x_1,\ldots,x_r\}\) are both linearly independent.)