Gautam Iyer
Office: WEH 6121.
Contacting me by email
For mathematical email queries other than minor clarifications or typos on the homework, I request you come speak to me in person instead of sending me an email. Mathematics is not easily expressed via email, and a physical in person conversation will be a lot more productive.
For minor clarifications, typos on the homework, and logistical queries, my email address is:
gi1242+341@NoSPAM.edu (replace "NoSPAM" with "cmu")
(Please get the numbers and the plus sign correct, as that will ensure that your email goes to my course folder.) I don't always check email in the evenings, so if you send me a desperate question about the homework at the 11th hour, then you're on your own.
Anonymous feedback
Feedback at any time (either anonymous or signed) is always appreciated. You can use this form to send me (or your course assistant) anonymous (or signed) feedback.
Note: Unfortunately, evil spammers have used this form to clutter my INBOX. Thus I have restricted access to this form to within cmu.edu domain. Any inconvenience caused is regretted.
LaTeX sources
LaTeX sources
If you're interested in typing your homework using LaTeX, you can find the LaTeX sources of all the homework here. Be sure you also download the calculus.sty style file.
WARNING: These files are provided as is, with no warranty whatsoever. The files all LaTeX cleanly, with no errors on my system. If they don't compile cleanly on your system, you're on your own.
Homework & Solutions
- HW1. (Solutions. Mean: 14.9)
- HW2. (Solutions. Mean: 18.3)
- HW3. (Solutions. Mean: 17.7)
- HW4. (Solutions. Mean: 17.8)
- HW5. (Solutions. Mean: 13.9)
- HW6. (Solutions. Mean: 13.0)
- HW7. (Solutions.)
- HW8. (Solutions.)
- HW9. (Solutions.)
- HW10. (Solutions.)
- HW11. (Solutions.)
- HW12. (Solutions.)
Handouts
- Lecture schedule (will continually be updated).
- 2009, fall course website
- A note on studying for exams
- 2009, fall: Midterm 1 & solutions.
- 2010, spring: Midterm 1.
- Brief, illegible notes on error correcting codes.
- Your Midterm 1 & solutions.
- 2009, fall: Midterm 2 & solutions.
- 2010, spring: Midterm 2.
- Your Midterm 2 & solutions.
- Previous finals: 2009 Fall and 2010 Spring. (Unfortunately solutions to these are not available.)
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SVD and compression
SVD and compression
As I mentioned in class, a natural application of the SVD is to find low rank approximations of matrices. Here's an example application to image compression. We take an image, treat it as an $m \times n$ matrix $A$, and compute it's SVD: $$ A = U \Sigma V^* $$ Now we retain only it's $r$ largest singular values. That is, let $\Sigma_r$ be the top left $r \times r$ entries of $\Sigma$, $U_r$ be the first $r$ columns of $U$, and $V_r$ the first $r$ columns of $V$. Then we use the matrix $$ A_r = U_r \Sigma_r V_r^* $$ as our rank-$r$ approximation for $A$.
Note that you originally needed $mn$ bytes to store the matrix $A$. However you only need $r(m + n + 1)$ bytes to store $A_r$ ($rm$ bytes for $U_r$, $r$ bytes for $\Sigma_r$, and $rn$ bytes for $V_r$). So, for example, if $m = n$, and you if only choose $p\%$ of the (largest) singular values of $A$, then you only need $2p\%$ of the original spaces to store the image.
Here are some results. (Click on the thumbnail for a bigger picture). The first five images retain the largest 5%, 10%, 15%, 20% and 30% of the singular values. (Recall, this means they compress the image to roughly 10%, 20%, 30%, 40% and 60% of the original size, respectively). The last image is the original image.
If you'd like to try this example yourself, here is the code I used to generate the files: svdimg.m. You'll need Octave to run it. If you're one of those who hasn't yet seen the light and insists on using proprietary software, you can probably port it back to Matlab code if you like.
- Your final and solutions
