University linear algebra test questions: Geometric meaning of singular vectors

📚 Quick Study Guide

• 📐 Singular Value Decomposition (SVD) decomposes a matrix $A$ into three matrices: $A = U \Sigma V^T$, where $U$ and $V$ are orthogonal matrices, and $\Sigma$ is a diagonal matrix with singular values.
• 🧭 The columns of $V$ (right singular vectors) are orthonormal vectors that form a basis for the domain of the linear transformation represented by $A$. They point in the directions of maximum stretching.
• 📈 The columns of $U$ (left singular vectors) are orthonormal vectors that form a basis for the range of the linear transformation. They represent the directions to which the corresponding right singular vectors are mapped after transformation.
• 🔢 The singular values in $\Sigma$ are the scaling factors (magnitudes of stretching) along the directions of the corresponding singular vectors. If $\sigma_i$ is a singular value, then $A\vec{v}_i = \sigma_i \vec{u}_i$, where $\vec{v}_i$ is a right singular vector and $\vec{u}_i$ is a left singular vector.
• 🧮 Geometrically, the right singular vectors are the principal axes of the hyperellipse (or hyperellipsoid in higher dimensions) that results from transforming the unit sphere by $A$.

Practice Quiz

1. What do the right singular vectors of a matrix $A$ represent geometrically?
   1. The directions of maximum compression.
   2. The eigenvectors of $A$.
   3. The principal axes of the hyperellipse resulting from transforming the unit sphere by $A$.
   4. The null space of $A$.


2. If $A = U\Sigma V^T$ is the SVD of $A$, what do the columns of $V$ represent?
   1. Left singular vectors.
   2. Right singular vectors.
   3. Eigenvectors of $A^TA$.
   4. Eigenvectors of $AA^T$.


3. The singular values in $\Sigma$ represent:
   1. Angles of rotation.
   2. Scaling factors along the directions of the singular vectors.
   3. Shifting factors.
   4. Shearing factors.


4. What is the geometric interpretation of applying a matrix $A$ to its right singular vector $\vec{v}_i$?
   1. It rotates $\vec{v}_i$ by 90 degrees.
   2. It projects $\vec{v}_i$ onto the null space of $A$.
   3. It scales $\vec{v}_i$ by a singular value and maps it to the corresponding left singular vector.
   4. It leaves $\vec{v}_i$ unchanged.


5. If a matrix $A$ transforms a unit sphere into a line segment, what can you say about its singular values?
   1. All singular values are equal to 1.
   2. One singular value is non-zero, and the rest are zero.
   3. All singular values are zero.
   4. The singular values alternate between 1 and -1.


6. Which of the following is NOT a property of singular vectors?
   1. They are orthonormal.
   2. They form a basis for the domain and range of the linear transformation.
   3. They are always eigenvectors of the matrix.
   4. They indicate the directions of maximum stretching.


7. What happens to the unit circle after being transformed by a 2x2 matrix A, with singular values 2 and 0.5?
   1. It remains a circle.
   2. It becomes a line segment.
   3. It becomes an ellipse with semi-major axis 2 and semi-minor axis 0.5.
   4. It becomes a square.