Solution: The matrix $\mathbf{M}$ is constructed by placing the images of the standard basis vectors as its columns. Thus, $\mathbf{M} = \begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}$. Verifying, $\mathbf{M} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, confirming correctness. $\boxed{\begin{pmatrix} 2 & 3 \\ -1 & 4 \end{pmatrix}}$

["Understanding the Matrix Construction from Standard Basis Images: A Fundamental Solution in Linear Algebra", "In linear algebra, one of the foundational concepts for representing linear transformations is constructing a matrix from the images of the standard basis vectors. This approach not only clarifies how a matrix encapsulates transformation behavior but also provides an elegant, coordinate-based understanding of linear maps.", "The matrix (\mathbf{M} = \begin{pmatrix} 2 & 3 \ -1 & 4 \end{pmatrix}) exemplifies this idea. Here, the matrix’s columns are formed by applying the respective standard basis vectors (\mathbf{e}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix}) and (\mathbf{e}_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix}) to the transformation defined by (\mathbf{M}).", "By definition:\n[\n\mathbf{M} = \begin{pmatrix} \mathbf{M}\mathbf{e}_1 & \mathbf{M}\mathbf{e}_2 \end{pmatrix}\n]", "Computing these:\n- (\mathbf{M} \begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} 2 \ -1 \end{pmatrix})\n- (\mathbf{M} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} 3 \ 4 \end{pmatrix})", "Thus, placing these results as columns gives:\n[\n\mathbf{M} = \begin{pmatrix} 2 & 3 \ -1 & 4 \end{pmatrix}\n]", "This construction confirms the matrix’s correctness since it faithfully reproduces how the transformation acts on each basis vector. As both vectors are linearly independent, (\mathbf{M}) represents a full-rank transformation, enabling efficient system modeling in applications like computer graphics, physics simulations, and data compression.", "Understanding this basis-mapping method deepens intuition about matrix representation and is essential for anyone studying or applying linear algebra in science, engineering, or machine learning. It bridges abstract concepts with practical computation, proving why constructing (\mathbf{M}) from standard images is both simple and powerful."]