MIT OCW 18.06SC Section: Elimination with Matrices
- Strang’s lecture: Elimination with Matrices
- Recitation with Martina Balagovic
- Reading: Strang 5th edition, Section 2.2 & 2.3
Supplementation:
- https://www.3blue1brown.com/
- Vectors
- Span
- Linear Transformations
- Matrix Multiplication
- 3D Transformations
- Determinant
- Inverse Matrices, Column Space, and Null Space
- Non-square Matrices
- Dot Products and Duality
- Schaum’s Outlines, Lipschutz & Lipson, 6th edition
- Chapters 1 – 3
- Many solved problems and supplementary problems
Hours Spent: ~30 hours
Summary
I started the Linear Algebra course more than two weeks ago. The first sections of the OCW, i.e. The Geometry of Linear Equations and An Overview of Key Ideas were very intuitive. The section on Elimination was also clear. However, understanding lecture and reading content is not the same as ability to calculate – I want to be able to apply the key concepts covered in a section confidently. For this, there is no substitute for doing the math.
Although Strang’s textbook does offer a number of practice exercises for reinforcing concepts covered in respective sections, it doesn’t provide a key for checking my answers. I was fortunate to find Schaum’s Outlines, an excellent supplement that covers a number of topics, briefly, and supports each with worked examples. At the end of units it provides solved problems (solutions with the step-by-step), as well as supplementary problems (just the solutions). This additional textbook was just what I needed.
So, after moving through the OCW lectures and reading, I dove into Schaum’s, filling in details between the lines outlined in OCW. This sent me down the rabbit hole of better understanding what vectors, matrics, linear combinations, and linear transformations are. I also found 3Blue1Brown, which offers spectacular educational videos on a variety of mathematical themes, including Linear Algebra. 3Blue1Brown emphasized seeing Linear Algebra as transformations in space, rather than just ordered lists of numbers operating on each other.
I took around two week to watch and rewatch 3Blue1Brown videos, read and re-read Strang’s and Schaum’s relevant sections (and then some), and practice many exercises. I feel like I’m now comfortable enough with the foundational ideas to move forward confidently. Here are just some key concepts getting going:
Vector
A vector is a list of numbers that represent a direction and a size. They are represented a s column of numbers. These are first depicted as arrows pointing somewhere in a 2D or 3D space. Where an arrow points is its direction, and the length of the arrow is its magnitude (size). For example, a car driving due North at 80 km/hour is a vector. We can add vectors together – for example, a boat travels in a river at 5 km/hour while the river pushes the boat laterally at 3 km/hour. We can also scale vectors (speed them up, slow them down, reverse them)
Linear Transformations
A linear transformation is a mapping of one vector to another, fixed to a single origin. Vectors can be stretched, shrunk, rotated, reflected, or sheared (like the tilting of a square sideways into a parallelogram). Similarly, a car can go faster or slower (stretch or shrink), its direction can turn (rotate), or it can drift to the left or right (shear). Linear transformations can be applied in sequence: one mapping followed by another. Each transformation keeps the origin fixed, and their combination is itself a linear transformation. The chaining of these linear transformations is matrix multiplication.
Matrix
A matrix is a collection of vectors arranged in rows or columns; it looks like a grid of numbers. Whereas a vector describes direction and size, a matrix describes how the vectors interact, or what happens to them. We use matrices to evaluate linear transformations.
Elimination
Elimination is a way of solving systems of equations using matrices. By scaling and adding rows, we eliminate variables step-by-step. Each row has fewer variables than the row preceding it. The last row has one variable, which we can solve for. After solving for the single variable, we can then substitute the variable into the preceding equation to find the value for the remaining unknown variable. Working all the way back to the top equation, we solve for (if possible) the variables of the collective equations. For example:
x – 2y =1
3x + 2y = 11
Here we can multiply the first equation by 3, then subtract it from the second equation, to wind up with
8y = 8, or y =1
With y solved, we can then easily solve x = 3.
Elimination can result in either one finite solution, or it otherwise fails when the lone variable equals 0.
Personal Observations
- At its core, linear algebra is algebra plus transformation.
- We use familiar algebraic operations to work with matrices.
- Matrices also describe how space itself is transformed, which we can visualize in 2D or 3D.
- Even when dimensions are too high to picture, we can still reason about how vectors are transformed in an n-dimensional space.
- Once you see vectors and transformations, you start seeing the same structure in code, physics, and economics.
Reflection
This is already a very interesting subject. So far it’s been much more theoretical than Calculus, which was quick to the mat to grind out problems. I have a more difficult time with the visualizing, geometrical element of math. 3Blue1Brown really challenged me to envision what is happening – excellent mental dead lifts. Additionally Throughout Schaum’s, we’re asked to prove theorems, something which is new to me entirely. I can’t wait to forge ahead.