From 8fa8665acbb6b6fa901404796f5c3dd2355a6804 Mon Sep 17 00:00:00 2001 From: Karan Jayachandra Date: Tue, 14 Apr 2026 21:07:54 +0200 Subject: Added the Linear Algebra draft --- content/posts/linear_algebra.md | 21 +++++++++++++++++++++ 1 file changed, 21 insertions(+) create mode 100644 content/posts/linear_algebra.md (limited to 'content/posts/linear_algebra.md') diff --git a/content/posts/linear_algebra.md b/content/posts/linear_algebra.md new file mode 100644 index 0000000..0be65f9 --- /dev/null +++ b/content/posts/linear_algebra.md @@ -0,0 +1,21 @@ ++++ +title = "Essence of Linear Algebra" +date = 2026-04-14 +math = true ++++ + +When I first encountered linear algebra in school, I failed to realize the profound impact it has on the world around us. At this moment all I could think of was that this was a tool to solve $n$ equations in $n$ unknowns. This recurred when I was studying Linear Algebra although at a higher level during my undergraduate studies. When studying for my master's degree I saw the gaping hole in knowledge I had in basic mathematics. Things my fellow students found intuitive would require me to spend time writing several equations. It was at this point that I realized that I needed a refresher course. At this moment I came across [Gilbert Strang's Lectures for MIT18.06](https://www.youtube.com/watch?v=7UJ4CFRGd-U&list=PLE7DDD91010BC51F8). This was a godsend. Without this course I do not think I would have my degree or a job. I even wrote an email to this effect to Prof. Strang who was also kind enough to respond and warm greetings. I find his [book](https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html) on this topic to be the definitive source of information on Linear Algebra. No article can hope to capture the effect that attending his lectures would. I highly recommend that you do it if you have the time. However if you are in need of a quick recap of its concept, I summarize my understanding of the course in a way that made sense. It is more of a reference for myself for the future when I might forget a thing or two. Here we go! + +## Vectors + +The fundamental unit of linear algebra is a vector. A vector is a collection of numbers that belong together. The most basic of which you might be familiar with as the cartesian coordinate system where the numbers represent the components along the $x$, $y$ and $z$ axes. + +$$ \mathbf{v} = \begin{bmatrix} x \\\\ y \\\\ z \end{bmatrix} $$ + +But it doesn't have to be this literal. A vector can contain any information. A more realistic example could be a vector containing id of a car dealership, the number of sedans sold at the location, the total number of sales persons and the number of years the location has been active. Any set of related information can be added to a vector. Even words can be used by creating a numerical representation of it. This is what is done by LLMs. A vector is not limited in dimensions. The examples above have 3 and 4 dimensions or in mathematical terms in $\mathcal{R}^3$ and $\mathcal{R}^4$. But this can be extended to any random number of dimensions, $\mathcal{R}^n$. Vectors can be added together and scale. A linear combination is a generalization of combining vectors in different amounts. Notice that vectors are usually denoted in bold in mathematical notation. + +$$ \mathbf{y} = c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2 + c_3 \mathbf{x}_3 $$ + +## Matrices + +A matrix is a useful way of representing a transformation. A matrix is a set of vector as described in the earlier section stacked together horizontally. Another way of thinking of a matrix is as a transformation of basis. In 3 dimensions, the unit vector $x$ axis is defined by $\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$ \ No newline at end of file -- cgit v1.3.1 From e0cc4edb874041bc3e5da7ea6fef978a7c9e5b5e Mon Sep 17 00:00:00 2001 From: Karan Jayachandra Date: Fri, 17 Apr 2026 22:32:52 +0200 Subject: Added the updates to the linear algebra post --- content/posts/linear_algebra.md | 14 +++++++- content/posts/tools.md | 72 +++++++++-------------------------------- 2 files changed, 28 insertions(+), 58 deletions(-) (limited to 'content/posts/linear_algebra.md') diff --git a/content/posts/linear_algebra.md b/content/posts/linear_algebra.md index 0be65f9..5239a97 100644 --- a/content/posts/linear_algebra.md +++ b/content/posts/linear_algebra.md @@ -18,4 +18,16 @@ $$ \mathbf{y} = c_1 \mathbf{x}_1 + c_2 \mathbf{x}_2 + c_3 \mathbf{x}_3 $$ ## Matrices -A matrix is a useful way of representing a transformation. A matrix is a set of vector as described in the earlier section stacked together horizontally. Another way of thinking of a matrix is as a transformation of basis. In 3 dimensions, the unit vector $x$ axis is defined by $\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$ \ No newline at end of file +A matrix is a useful way of representing a transformation. A matrix is a set of vector as described in the earlier section stacked together horizontally. Another way of thinking of a matrix is the transformation of reference axes. In 3 dimensions, the unit vector along the $x$, $y$ and $z$ axis is defined as $\begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$, $\begin{bmatrix} 0 & 1 & 0 \end{bmatrix}$ and $\begin{bmatrix} 0 & 0 & 1 \end{bmatrix}$. If we stack these one top of each other, we arrive at the identity matrix or the default axes. We can transform these axes to point to any other directions by just stacking those vectors into a matrix and then multiplying them. Notice that the operation carried out is now just the dot product of the vector with axes that was just defined. The dot product is just a measure of how much the vector is pointing in the direction of another vector which in our case is the new axes. + +$$ \mathbf{v} = \begin{bmatrix} 1 & 2 & 3\\\\ 3 & 5 & 8 \\\\ 11 & 3 & 1 \end{bmatrix} \begin{bmatrix} x \\\\ y \\\\ z \end{bmatrix} = \begin{bmatrix} x + 2y + 3z \\\\ 3x + 5y + 8z \\\\ 11x + 3y + z \end{bmatrix} $$ + +This operation is another way of representing the linear transformation discussed in the previous section. I like to this of this operation as refocusing any and all vectors to a different section of space. This can be generalized with a change in the origin as well using the affine transformation written as: + +$$ \mathbf{y} = \mathbf{A} \mathbf{x} + \mathbf{b} $$ + +where $\mathbf{A}$ is the change in perspective and $\mathbf{b}$ is the change in point of reference. + +## Matrix Rank + +This section is to be continued. \ No newline at end of file diff --git a/content/posts/tools.md b/content/posts/tools.md index c22dfbb..56b5d1b 100644 --- a/content/posts/tools.md +++ b/content/posts/tools.md @@ -1,69 +1,27 @@ +++ -title = "Tools I endorse" +title = "My toolkit" date = 2026-03-25 +++ -Computing is here to stay and can be used to enhance or take away from life. I am also biased towards change. +Technology has transformed our lives. However, sometimes it overextend their reach into certain areas of my life. There I draw clear boundaries on what my use cases are where computers, phones and other general purpose devices can help me. These are listed below in the order of priority. -It is also part of my life in a way enhances and drains from it. I am a technology addict and I find it fun to tweak things here and there all the me time. The longer I am struggling with this, the longer it takes for me to work on things that matter to me. +1. Communication +2. Services +3. Documents +4. Engineering +5. Media +6. Time Management -Using technology brings problems as well which I need to manage, such as security, ads and passwords. I use Adguard VPN and Bitwarden for each of these respectively. +I have a phone and a backup along with a laptop and a tablet which acts as its back to achieve these use cases. Before using any of these though, I recommend everyone to setup a password manager. I use [Bitwarden](https://bitwarden.com). Use an ad blocker as well. This will reduce distractions when browsing the internet. I recommend [uBlock](https://ublockorigin.com). Any system I create: -Therefore, here I am making a list of things that technology actually helps me with in the order of importance to me. This can guide me in what exactly I need: +## Must haves -1. Communication: Calls, texts and emails -2. Knowledge Management: Text and number crunching -3. Engineering tools: System design and programming -4. Media consumption: Audio, images, video, and gaming -5. Accessing services: Navigation, banking and shopping -6. Time Management: Calendar and reminders +I _communicate_ with family and friends via text messages and calls. I prefer using Signal for messaging but also use Whatsapp. I also have an email attached to my domain and a separate email address which I fill in when I don't want to provide my primary one. I believe that email is a better form of communication that instant messaging aas it respects the time of the correspondents. Many services in the modern world are also becoming more digital. Governmental agencies, banking, navigation or online only stores are also important _services_ that I need to access. -Based on the requirements I have, I can make a list of software that I need. +## Should haves -## Communication +Managing documents is much easier with a digital device. Once digitized they are always available and can be backed up. I use a personal network attached storage (NAS) and iCloud for this. I also cannot live without the use of engineering tools that have made life much easier. These are tools for programming, hardware and mechanical design. If you want to learn something from scratch, I highly recommend Python, KiCAD and OpenSCAD respectively. I also use typesetting tools such as LaTeX or Typst to generate high quality PDFs to share with others. [Ipe](https://ipe.otfried.org) is also a great tool to augment the others for diagrams. I try to stay away from tools that could take my data hostage via proprietary formats or other forms of lock in. I use plain text via Markdown when possible. I prefer to use VS Code for programming or NeoVim for text documents. I also generate HTML using hugo to crates websites such as this to store information that I share with others. I have digital music and images that I also store on iCloud and my NAS so that I can stream from it to many devices. I keep this locked behind a Tailnet to precent unauthorized access. -For communication, I use calls and texts via a phone along with internet versions of this via Signal. For email, I use my own domain name. I can be reached at me@karanj.com for most purposes but I have a throw away email to subscribe for newsletters and for services that I don’t trust. I need at least the calls accessible at all times for emergencies so it need to be part of my everyday cary. +## Could haves -## Time Management and Services - -For reminders and calendars, I am not picky, anything goes. I don’t care for retaining it perpetually as well so it doesn’t need to be backed up. It does need to be accessible to me all the time, it has to be part of my everyday cary. Certain services such as banking and navigation also need to be available at all time. Other services such as shopping is less important. For shopping, I pay for Amazon Prime as it saves me money overall when ordering things. - -## Images - -I want to be able to store photos and search through them. I want to use digital frames to display photos in the house. But I do care about photos and need to find a way to back things up to two separate drive. In this case personal videos are also considered in the image category. It helps to have this also all the time to show others but isn’t necessary. - -## Gaming -I am not a gamer, I just enjoy games. This means that a dedicated PC isn’t worth it for me. Gaming is also more convenient with a console. I don’t care much about games either, so I don’t want to back up or buy physical copies of games. I also don’t need this all the time so this can be at home. - -## Videos -I want to download movies and TV shows but store only the ones that really matter to me. These items should be available specifically at home and having this with me is a bonus. I don’t need this to be backed up too much. Maybe only my favourite movies. When possible I use VLC to watch videos as I like the story behind it and it is open-source. I also pay for YouTube because I find the service worth it. But I am not a fan of Google so want to minimise the amount I services I use them for because: - -* Google maps destroys restaurants. - -## Audio - -I use Apple Music to listen to music because of their extensive catalogue and better pay for musicians compared to Spotify. But this isn’t important to me. I can switch services at any time. But music matters to me so I do want to make physical copies that I have at home. - -## Engineering Tools - -I use specialized software for: - -- Programming -- Design: OpenSCAD and KiCAD - -## Knowledge Management - -Managing information is the most important part of my life. I care a lot about intellectual growth and I need this part of my life to work smoothly. I can use programming to crunch numbers. - -### Obsidian - -Plain text is [unreasonably effective](https://www.youtube.com/watch?v=WgV6M1LyfNY) in organizing information. Markdown adds a bit of syntax to keep things cleaner and easier to read. Obsidian uses markdown and adds the functionality to link articles. Things cannot get better than this. It also helps study and revision which is extremely important to retain information and recall when needed. It is accessible across all platforms and works on more or less any device. I pay for their sync feature because of how seamless it is. I also back this up to two separate drives. The same ones used for photos. -I tried using plain text documents via git and also Apple Notes and struggled with both of them. The first had issues with accessibility on my phone and the second had horrible user experience. - -### Typst / LaTeX - -Typst is a wonderful effort to reimagine LaTeX for the modern world. I can create beautiful looking CVs, letters, reports, books and presentation. The most popular option and the one used by journal however is LaTeX. I therefore need to have this available. - -### Ipe - -[Ipe](https://ipe.otfried.org) creates amazing block diagrams. When we have a complicated diagram, the basic packages for both Typst and LaTeX don’t cut it. Follow this [guide](https://algo.win.tue.nl/tutorials/ipe/). It allows for automatic conversion to other formats on saves by changes the [prefs.lua](https://mailman.science.uu.nl/pipermail/ipe-discuss/2011-July/001344.html) and setting the [auto export](https://mailman.science.uu.nl/pipermail/ipe-discuss/2011-July/001342.html). There is also a useful utility that can [batch convert](https://otfried.github.io/ipe/94_commandline_programs.html#ipetoipe-converting-ipe-file-formats) all your .ipe files. \ No newline at end of file +I am not picky with reminders and a calendar. I use iCloud for now as I don't mind loosing this information. I also store movies and TV shows on my NAS which allows me to consume and own my media collection. I occasionally game for which I use a Playstation 5. \ No newline at end of file -- cgit v1.3.1