{%hackmd 5xqeIJ7VRCGBfLtfMi0_IQ %} # Final exam: Questions to ponder Please see [Course website](https://www.math.nsysu.edu.tw/~chlin/2024SMath104A/2024SMath104A.html) for details about the exam. :::success :bulb: You need to provide the reasons for your answers. ::: ## How to prepare the exam? 1. Make a **concrete setting** for the problems whenever possible. For example, what is the length of $(1,2,3)$? 2. **Write down** your answer and reasons on a paper. 3. If you do not know how to answer it, **look it up** from the following resources, or ask ChatGPT. - `Hefferon` [_Linear Algebra_](http://joshua.smcvt.edu/linearalgebra/book.pdf) by Jim Hefferon - `NB` [_Linear algebra notebook_](https://jephianlin.github.io/LA-notebook/index-en.html) by Jephian Lin - `LR` [Learning resources](https://hackmd.io/@jephianlin/2024SMath104A-resources) 5. **Think carefully** whether your answer is correct or not. 6. Repeat the above steps with **different settings**. ## Matrix representation - Describe the meaning of the matrix representation $[f]_\alpha^\beta$. - For any $m\times n$ matrix, explain why $A = [f_A]_{\mathcal{E}_n}^{\mathcal{E}_m}$, where $\mathcal{E}_k$ is the standard basis of $\mathbb{R}^k$. - Explain why $[\idmap]_\beta^{\mathcal{E}_n}$ is the matrix whose columns are vectors in $\beta$. - Given $[\idmap]_\beta^{\mathcal{E}_n}$, how to find $[\idmap]_{\mathcal{E}_n}^\beta$? - Given an $m\times n$ matrix $A$, a basis $\alpha$ of $\mathbb{R}^n$, and a basis $\beta$ of $\mathbb{R}^m$, how to find $[f_A]_\alpha^\beta$? Related resources: [LR of Math103A](https://hackmd.io/@jephianlin/2023FMath103A-resources) 31~35 ## Projection - Given a subspace $V = \vspan\{\bu\}$ and a vector $\bb$, find the projection of $\bb$ onto $V$. - Given a subspace $V = \vspan\{\bu_1, \ldots, \bu_d\}$ and a vector $\bb$, find the projection of $\bb$ onto $V$. - Given a matrix $A$ and a vector $\bb$, find $\bx$ that minimizes $\|A\bx - \bb\|^2$. - Given $\bx = (x_1, \ldots, x_N)\trans$ and $\by = (y_1, \ldots, y_N)\trans$, find $c_0$ and $c_1$ such that $f(x) = c_0 + c_1x$ minimizes $\sum_{i=1}^N (f(x_i) - y_i)^2$. Related resources: `Hefferon Three.VI`, `LR 29~33`, `NB 105` ## Spectral theorm - Define an orthogonal basis and an orthonormal basis. - Define an orthogonal matrix. - Given a matrix $A$, how to find an orthonormal basis of $\ker(A)$? - Describe the spectral theorem. - Given a real symmetric matrix $A$, find an orthonormal basis composed of eigenvectors that diagonalizes $A$. - Given a real symmetric matrix $A$, find the spectral decomposition of $A$. - Describe the meaning of the spectral decomposition. Related resources: `LR 34~36`, `NB 603, 604` ## Rayleigh quotient - Define Rayleigh quotient. - Describe the Rayleigh quotient theorem. - Given a diagonal matrix $A$, find the maximum Rayleigh quotient value and the vector that achieves it. - Given a real symmetric matrix $A$, find the maximum Rayleigh quotient value and the vector that achieves it. Related resources: `LR 37~38`, `NB 609` ## Singular value decomposition - Describe the singular value decomposition. - Describe the meaning of the singular value decomposition. - Explain why $AA\trans$ and $A\trans A$ are symmetric. - Explain why $AB$ and $BA$ have the same nonzero eigenvalues. - What is the relations between the eigenvalues of $AA\trans$, or $A\trans A$, and the singular value of $A$? Related resources: `LR 39~41`, `NB 314, 605` *This note can be found at Course website > Learning resources.*