📚 Topic Summary
The Singular Value Decomposition (SVD) is a factorization of a real or complex matrix. Specifically, for a matrix $A$, the SVD is a decomposition of the form $A = U \Sigma V^T$, where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal matrix containing the singular values of $A$. SVD provides critical insight into the fundamental subspaces associated with a matrix, including the range (column space), null space (kernel), row space, and left null space. Understanding the relationship between SVD and these subspaces is fundamental in many applications of linear algebra.
This activity sheet explores how the SVD reveals and defines these fundamental subspaces.
🧠 Part A: Vocabulary
Match the term with its correct definition.
| Term |
Definition |
| 1. Singular Value |
A. The set of all vectors that, when multiplied by the matrix, result in the zero vector. |
| 2. Null Space |
B. A matrix whose columns are orthonormal eigenvectors of $A^TA$. |
| 3. Column Space |
C. The square root of the eigenvalues of $A^TA$. |
| 4. Left Null Space |
D. The set of all linear combinations of the columns of $A$. |
| 5. Right Singular Vectors |
E. The set of all vectors that, when multiplied by $A^T$, result in the zero vector. |
✏️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
The SVD of a matrix $A$ is given by $A = U \Sigma V^T$. Here, $U$ contains the _______ singular vectors, $V$ contains the _______ singular vectors, and $\Sigma$ is a _______ matrix containing the singular _______. The columns of $U$ corresponding to non-zero singular values span the _______ of $A$, while the columns of $V$ corresponding to non-zero singular values span the _______ of $A$.
🤔 Part C: Critical Thinking
Explain how the SVD can be used to find the best rank-k approximation of a matrix $A$, and why this is useful in applications like data compression or noise reduction.