Solution: The columns of $ \mathbf{M} $ are the images of the standard basis vectors. Thus, $ \mathbf{M} = \begin{bmatrix} 3 & 1 \\ -2 & 4 \end{bmatrix} $. Verification:

["# Understanding the Matrix $ \mathbf{M} $: Columns as Images of Standard Basis Vectors", "In linear algebra, one of the foundational concepts is how matrices represent linear transformations by mapping vectors, including the standard basis vectors. This article explores the structure of the matrix\n$$\n\mathbf{M} = \begin{bmatrix} 3 & 1 \ -2 & 4 \end{bmatrix}\n$$\nand demonstrates—via rigorous verification—that its columns indeed represent the images of the standard basis vectors under the transformation defined by $ \mathbf{M} $.", "---", "## What Are the Standard Basis Vectors?", "In $ \mathbb{R}^2 $, the standard basis consists of two unit vectors:\n- $ \mathbf{e}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix} $\n- $ \mathbf{e}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix} $", "These vectors define the coordinate system used to describe all vectors in $ \mathbb{R}^2 $.", "---", "## The Matrix Representation of a Linear Transformation", "A $ 2 \ imes 2 $ matrix $ \mathbf{M} $ defines a linear transformation $ T: \mathbb{R}^2 \ o \mathbb{R}^2 $ such that for any vector $ \mathbf{v} = \begin{bmatrix} x \ y \end{bmatrix} $,\n$$\nT(\mathbf{v}) = \mathbf{M} \mathbf{v}\n$$", "But more importantly, the columns of $ \mathbf{M} $ are precisely the images of the standard basis vectors under $ T $. That is:\n- $ T(\mathbf{e}_1) = \ ext{first column of } \mathbf{M} $\n- $ T(\mathbf{e}_2) = \ ext{second column of } \mathbf{M} $", "---", "## Verifying the Columns of $ \mathbf{M} = \begin{bmatrix} 3 & 1 \ -2 & 4 \end{bmatrix} $", "Let’s compute $ \mathbf{M} \mathbf{e}_1 $ and $ \mathbf{M} \mathbf{e}_2 $ to confirm these are the column vectors.", "### Step 1: Compute $ T(\mathbf{e}_1) = \mathbf{M} \mathbf{e}_1 $", "$$\n\mathbf{e}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\n\quad \Rightarrow \quad\n\mathbf{M} \mathbf{e}_1 = \begin{bmatrix} 3 & 1 \ -2 & 4 \end{bmatrix} \begin{bmatrix} 1 \ 0 \end{bmatrix} = \begin{bmatrix} 3 \cdot 1 + 1 \cdot 0 \ -2 \cdot 1 + 4 \cdot 0 \end{bmatrix} = \begin{bmatrix} 3 \ -2 \end{bmatrix}\n$$", "This matches the first column of $ \mathbf{M} $. So, $ \mathbf{M} \mathbf{e}_1 $ correctly represents the image of $ \mathbf{e}_1 $ under $ T $.", "### Step 2: Compute $ T(\mathbf{e}_2) = \mathbf{M} \mathbf{e}_2 $", "$$\n\mathbf{e}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\n\quad \Rightarrow \quad\n\mathbf{M} \mathbf{e}_2 = \begin{bmatrix} 3 & 1 \ -2 & 4 \end{bmatrix} \begin{bmatrix} 0 \ 1 \end{bmatrix} = \begin{bmatrix} 3 \cdot 0 + 1 \cdot 1 \ -2 \cdot 0 + 4 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 \ 4 \end{bmatrix}\n$$", "This matches the second column of $ \mathbf{M} $. Therefore, $ \mathbf{M} \mathbf{e}_2 $ gives the image of $ \mathbf{e}_2 $.", "---", "## Conclusion: A Direct Proof via Basis Imaging", "By construction, every column of a matrix $ \mathbf{M} $ in standard basis is the result of applying $ \mathbf{M} $ to the corresponding standard basis vector:\n$$\n\mathbf{M} \mathbf{e}_1 = \ ext{first column}, \quad \mathbf{M} \mathbf{e}_2 = \ ext{second column}\n$$\nSince\n$$\n\mathbf{M} = \begin{bmatrix} 3 & 1 \ -2 & 4 \end{bmatrix},\n$$\nit follows verifiably that the columns are the images of $ \mathbf{e}_1 $ and $ \mathbf{e}_2 $ under the transformation defined by $ \mathbf{M} $.", "---", "## Why This Matters", "Understanding that matrix columns are the images of standard basis vectors unlocks key insights into linear transformations, change of basis, and coordinate representations. It shows that matrices are not just abstract arrays but geometric tools mapping space via basis-efficient rules.", "By verifying columns this way, students and practitioners confirm that matrix definitions are consistent with linear algebra theory.", "---", "Key Takeaway:\nThe columns of $ \mathbf{M} = \begin{bmatrix} 3 & 1 \ -2 & 4 \end{bmatrix} $ are indeed $ \begin{bmatrix} 3 \ -2 \end{bmatrix} $ and $ \begin{bmatrix} 1 \ 4 \end{bmatrix} $, visualized as $ T(\mathbf{e}_1) $ and $ T(\mathbf{e}_2) $, respectively—dense validation of matrix-basis relationships.", "---", "## Further Reading", "- Understand linear transformations and basis change\n- Explore how determinants relate to scaling in basis images\n- Study change of basis matrices using standard basis mappings", "---", "By anchoring theory to concrete computation, this verification solidifies a cornerstone of linear algebra: matrices as matrices of images under transformation."]