Question: Find the matrix $ \mathbf{M} $ such that $ \mathbf{M} egin{pmatrix} x \ y \end{pmatrix} = egin{pmatrix} -y \ x \end{pmatrix} $.
["# How to Find the Matrix $ \mathbf{M} $ That Swaps $ x $ and $ y $ as $ \begin{pmatrix} -y \ x \end{pmatrix} $", "When working with linear transformations in linear algebra, one common problem is determining the matrix $ \mathbf{M} $ that transforms a vector $ \begin{pmatrix} x \ y \end{pmatrix} $ into $ \begin{pmatrix} -y \ x \end{pmatrix} $. This particular transformation swaps the $ x $ and $ y $ coordinates and negates the new $ x $ component — a rotation by $ 90^\circ $ counterclockwise combined with a sign flip.", "In this article, we’ll clearly explain how to find matrix $ \mathbf{M} $, understand the underlying linear transformation, and provide step-by-step guidance on computing $ \mathbf{M} $.", "---", "## Understanding the Transformation", "We are given a linear mapping defined by:\n[\n\mathbf{M} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -y \ x \end{pmatrix}\n]", "This transformation maps each input vector $ \begin{pmatrix} x \ y \end{pmatrix} $ to a new vector where:\n- The first component is $ -y $\n- The second component is $ x $", "This corresponds to a rotation by $ 90^\circ $ counterclockwise followed by a coordinate reflection.", "However, instead of focusing only on geometric interpretation, let’s determine the matrix $ \mathbf{M} $ directly using the standard basis vectors.", "---", "## Step-by-Step Derivation of $ \mathbf{M} $", "Let $ \mathbf{e}_1 = \begin{pmatrix} 1 \ 0 \end{pmatrix} $, $ \mathbf{e}_2 = \begin{pmatrix} 0 \ 1 \end{pmatrix} $ be the standard basis vectors.", "Apply $ \mathbf{M} $ to $ \mathbf{e}_1 $:\n[\n\mathbf{M} \begin{pmatrix} 1 \ 0 \end{pmatrix} = \begin{pmatrix} -0 \ 1 \end{pmatrix} = \begin{pmatrix} 0 \ 1 \end{pmatrix}\n]", "Apply $ \mathbf{M} $ to $ \mathbf{e}_2 $:\n[\n\mathbf{M} \begin{pmatrix} 0 \ 1 \end{pmatrix} = \begin{pmatrix} -1 \ 0 \end{pmatrix}\n]", "Now, construct $ \mathbf{M} $ using these results as columns:\n[\n\mathbf{M} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}\n]", "---", "## Why This Works", "Let’s verify the result:\n[\n\mathbf{M} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 0 \cdot x + (-1) \cdot y \ 1 \cdot x + 0 \cdot y \end{pmatrix} = \begin{pmatrix} -y \ x \end{pmatrix}\n]", "The computation confirms that the matrix $ \mathbf{M} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} $ performs the desired transformation.", "---", "## Mathematical Interpretation: Rotation Matrix", "The matrix $ \mathbf{M} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} $ is a well-known anti-clockwise rotation matrix. Specifically, it rotates vectors by $ \frac{\pi}{2} $ (90 degrees) in the plane.", "This explains why applying this matrix to $ \begin{pmatrix} x \ y \end{pmatrix} $ yields a vector with components swapped and negated — that’s exactly how rotation by $ 90^\circ $ transforms coordinates.", "---", "## Applications and Significance", "This transformation appears in many areas, including:\n- Computer graphics: rotating points in 2D space\n- Signal processing: phase-shift transformations\n- Geometry: coordinate changes under orientation-preserving rotations", "Understanding $ \mathbf{M} $ helps build intuition for more complex linear operators and transformations.", "---", "## Conclusion", "To find matrix $ \mathbf{M} $ such that\n[\n\mathbf{M} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} -y \ x \end{pmatrix},\n]\nyou determine how $ \mathbf{M} $ acts on the standard basis vectors and place those outputs as columns.", "The result is:\n[\n\boxed{ \mathbf{M} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} }\n]", "This matrix represents a $ 90^\circ $ counterclockwise rotation combined with reflection, making it fundamental in linear algebra and its applications.", "---", "## Additional Keywords for SEO Optimization", "- Matrix transformation\n- Linear algebra matrix derivation\n- Rotating matrix 2D\n- Coordinate transformation matrix\n- Find M such that $ \mathbf{M} \mathbf{v} = \begin{pmatrix} -y \ x \end{pmatrix} $\n- 2D vector transformation\n- Linear operator matrix\n- 90 degree rotation matrix", "Using these terms naturally enhances visibility in search engines for learners and educators exploring matrix operations and linear transformations.", "---", "If you’re studying linear algebra or beginning programming with vector transformations, mastering $ \mathbf{M} = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix} $ is essential — it encapsulates rotation and sign-flip behavior in a compact, beautiful mathematical form."]
