\mathbf{M} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}, \quad \mathbf{M} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}

["# Understanding the Linear Transformation Defined by Matrix M Given Two Vectors", "When faced with vector equations involving a matrix, one of the most insightful problems is analyzing how the matrix transforms standard basis vectors. Consider the system:", "[\n\mathbf{M} \begin{pmatrix} 1 \ 2 \end{pmatrix} = \begin{pmatrix} 3 \ 4 \end{pmatrix}, \quad \mathbf{M} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} 1 \ 1 \end{pmatrix}\n]", "These equations define a linear transformation represented by a $2 \ imes 2$ matrix ( \mathbf{M} ). By leveraging the linearity of matrix multiplication, we can reconstruct ( \mathbf{M} ) explicitly and uncover deeper insights into its behavior.", "---", "## Step 1: Express Standard Basis Vectors in Terms of Input Vectors", "The vectors ( \begin{pmatrix} 1 \ 2 \end{pmatrix} ) and ( \begin{pmatrix} 0 \ 1 \end{pmatrix} ) are not standard basis vectors but are linearly independent and span ( \mathbb{R}^2 ). To find ( \mathbf{M} ), we need to determine how it transforms the standard basis vectors:", "[\n\mathbf{e}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \quad \mathbf{e}_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix}\n]", "We already know:\n[\n\mathbf{M} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} 1 \ 1 \end{pmatrix}\n]", "Now, express ( \mathbf{e}_1 ) as a linear combination of the known input vectors:\n[\n\begin{pmatrix} 1 \ 0 \end{pmatrix} = a \cdot \begin{pmatrix} 1 \ 2 \end{pmatrix} + b \cdot \begin{pmatrix} 0 \ 1 \end{pmatrix}\n= \begin{pmatrix} a \ 2a + b \end{pmatrix}\n]", "Matching components:\n[\na = 1, \quad 2a + b = 0 \Rightarrow 2(1) + b = 0 \Rightarrow b = -2\n]", "Thus,\n[\n\begin{pmatrix} 1 \ 0 \end{pmatrix} = 1 \cdot \begin{pmatrix} 1 \ 2 \end{pmatrix} - 2 \cdot \begin{pmatrix} 0 \ 1 \end{pmatrix}\n]", "Using linearity:\n[\n\mathbf{M} \begin{pmatrix} 1 \ 0 \end{pmatrix} = 1 \cdot \mathbf{M} \begin{pmatrix} 1 \ 2 \end{pmatrix} - 2 \cdot \mathbf{M} \begin{pmatrix} 0 \ 1 \end{pmatrix} = 1 \cdot \begin{pmatrix} 3 \ 4 \end{pmatrix} - 2 \cdot \begin{pmatrix} 1 \ 1 \end{pmatrix}\n]", "Compute:\n[\n\begin{pmatrix} 3 \ 4 \end{pmatrix} - \begin{pmatrix} 2 \ 2 \end{pmatrix} = \begin{pmatrix} 1 \ 2 \end{pmatrix}\n]", "Hence,\n[\n\mathbf{M} \begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} 1 \ 2 \end{pmatrix}\n]", "---", "## Step 2: Write Matrix M Using Basis Transformation", "Now that we know how ( \mathbf{M} ) acts on ( \mathbf{e}_1 ) and ( \mathbf{e}_2 ), we can write ( \mathbf{M} ) explicitly. The transformation matrix columns are the images of the standard basis vectors:", "[\n\mathbf{M} = \begin{pmatrix} \mathbf{M} \mathbf{e}_1 & \mathbf{M} \mathbf{e}_2 \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix}\n]", "---", "## Step 3: Verify Consistency with Given Equations", "Check both original equations:", "1.\n[\n\mathbf{M} \begin{pmatrix} 1 \ 2 \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix} \begin{pmatrix} 1 \ 2 \end{pmatrix} = \begin{pmatrix} 1\cdot1 + 1\cdot2 \ 2\cdot1 + 1\cdot2 \end{pmatrix} = \begin{pmatrix} 3 \ 4 \end{pmatrix} \quad \checkmark\n]", "2.\n[\n\mathbf{M} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} 0 + 1 \ 0 + 1 \end{pmatrix} = \begin{pmatrix} 1 \ 1 \end{pmatrix} \quad \checkmark\n]", "Both match, confirming correctness.", "---", "## Step 4: Properties and Interpretation", "- Matrix Representation:\n[\n\mathbf{M} = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix}\n]", "- Determinant and Invertibility:\n[\n\det(\mathbf{M}) = (1)(1) - (1)(2) = 1 - 2 = -1\n]", "A nonzero determinant confirms ( \mathbf{M} ) is invertible. The matrix preserves area but reverses orientation (due to negative determinant).", "- Eigenvalues and Eigenvectors:\nSolve ( \det(\mathbf{M} - \lambda \mathbf{I}) = 0 ):", "[\n\begin{vmatrix} 1 - \lambda & 1 \ 2 & 1 - \lambda \end{vmatrix} = (1 - \lambda)^2 - 2 = \lambda^2 - 2\lambda - 1 = 0\n]", "Solutions:\n[\n\lambda = 1 \pm \sqrt{2}\n]", "These eigenvalues indicate stretching and contraction along eigen-directions.", "---", "## Conclusion", "By expressing the input vectors as linear combinations of standard basis vectors and using linearity, we reconstructed the transformation matrix ( \mathbf{M} ) as:", "[\n\boxed{ \mathbf{M} = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix} }\n]", "This matrix maps input vectors through linear combinations, preserving structured relationships between inputs and outputs. Understanding such transformations is fundamental in linear algebra, with applications in computer graphics, physics, engineering, and data science.", "Whether solving systems, analyzing geometric transformations, or modeling real-world phenomena, knowing how matrices act on known vectors unlocks powerful problem-solving tools.", "---", "### Key Search Terms\n- Matrix transformation from vector inputs\n- Finding matrix from vector equations\n- Linear algebra: define matrix from base vectors\n- How to compute M when M(e₁) and M(e₂) are known\n- Solving for transformation matrix using linearity\n- Linear transformations and basis vectors", "---", "Learn more about matrix transformations, eigenvalues, and applications in linear algebra textbooks or online courses tailored to advanced vector space theory."]