Lecture notes · Linear Algebra · Lecture 1

The geometry of linear equations

MIT 18.06 Linear Algebra, lecture 1, explained in plain English

Want this lecture to stick? In LessonBound you get checked practice questions for every part, can ask about anything you don't get, and can listen with your screen off. Free, no ads.

Learn this lecture free

Watch or read the original lecture: MIT OpenCourseWare.

Part 1 · 0:00–10:00

This part introduces the main problem of linear algebra, solving a set of linear equations, and shows it three ways: as lines, as a matrix, and as columns.

Key ideas

The main problem is to solve a set of linear equations. The lecture starts with the easy case where there are as many equations as unknowns. 0:55
A matrix is a rectangular grid of numbers. For the example, the grid holds the numbers that multiply x and y, and the unknowns x and y go into a vector. 3:09
The row picture draws each equation as a straight line. Every point on a line solves that equation, and the answer is the point where the lines meet. 5:42
The point x = 1, y = 2 lies on both lines, so it solves both equations. Plugging it in confirms this. 7:36
The column picture asks you to find the right mix of the matrix's columns that adds up to the right-hand side. The lecture calls this a linear combination and says it is the most basic operation in the course. 9:29

Words to know

matrix: A rectangular box of numbers arranged in rows and columns.
vector: A list of numbers, such as x and y, that goes together as one unit.
coefficient matrix: The matrix made from the numbers that multiply the unknowns in the equations.
linear combination: A total made by multiplying some things by chosen amounts and adding them up.

Example

Think of mixing paint. If you have a blue can and a yellow can, you can pour some of each to make a certain shade of green. A linear combination works the same way: you choose how much of each column to use so the mix comes out to the target. The lecture does not give this example; it is only an illustration.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part shows that solving linear equations can be pictured as combining vectors (arrows) and as finding where planes meet.

Key ideas

A linear combination means multiplying each column (arrow) by a number and adding the results together. The goal is to find the numbers that land on the target point b. 10:01
Picture the combination by placing the arrows tip to tail. Taking one of column one and two of column two lands exactly on the target (0, 3). 11:53
If you try all possible combinations of the two columns, you can reach any right-hand side. For two columns in a flat plane, they fill the whole plane. 14:35
With three equations and three unknowns, the row picture shows each equation as a flat plane in 3D space. Each linear equation's solutions form a plane. 19:00
A three by three matrix holds the coefficients of the three equations, and the right-hand side is a separate vector. Writing it this way is a short form for the full set of equations. 16:22

Words to know

Linear combination: A total you get by multiplying some numbers by vectors and adding them up.
Vector: An arrow that has a length and direction, described by a list of numbers such as (2, -1).
Matrix: A neat grid of numbers, here holding the coefficients of the equations, written in rows and columns.
Plane: A flat surface that goes on forever, like a sheet of paper that extends in every direction.

Example

Think of mixing paint from two colors of paint tubes. Using some amount of each tube and stirring them together is like a linear combination: you choose how much of each to add, and the result is one mixed color. With enough choices of amounts, you can reach many different colors.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how three equations can be seen as three planes meeting at one point, and how the same answer can be seen by mixing the columns of the matrix.

Key ideas

Each row of a three-by-three set of equations describes a flat surface called a plane. Two planes meet in a line, and three planes can meet in a single point. 20:00
The lecturer switches to the column picture. Here the solution is the mix of the matrix's columns that adds up to the right-hand side. 22:14
In this example the right-hand side is exactly one of the columns, so the answer is easy to see: use column three and none of the others. 24:33
The bigger question is whether you can solve the equations for every possible right-hand side. In column words, can mixes of the columns fill all of three-dimensional space? 27:55
For this matrix, the lecturer says the answer is yes. Elimination, the method in the next lecture, is how software finds the answer in general. 29:38

Words to know

plane: A flat surface that goes on forever, like a sheet of paper that never ends. In three dimensions, each equation with three unknowns gives one.
linear combination: A recipe that scales some vectors (lists of numbers) and adds them together. Here it means multiplying each column by a number and adding the results.
column picture: A way of looking at the equations by treating the matrix's columns as vectors and asking which mix of them makes the right-hand side.
elimination: A step-by-step method for solving a set of equations systematically. The lecture says it is the method that software uses.

Example

Think of mixing paint from three base colors. Each base color is a column, and the amounts you pour in are the numbers x, y and z. Asking whether you can make a target color is like asking whether a combination of the columns equals b.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–39:29

This part explains when a matrix can reach every right-hand side b, and how multiplying a matrix by a vector means mixing its columns.

Key ideas

A matrix is called invertible (or non-singular) when its columns let you reach every right-hand side b. The lesson says these are the matrices the course likes best. 30:19
If all the columns lie in the same plane, their combinations stay in that plane. Then only some right-hand sides b can be reached, so the matrix is singular and has no solution for every b. 31:04
In nine dimensions, if the ninth column is the same as the eighth, the combinations fill only an eight-dimensional plane inside nine-dimensional space, so some b cannot be reached. 34:45
Multiplying a matrix A by a vector x can be thought of as a linear combination of A's columns: take some amount of each column and add them up. 38:36

Words to know

invertible: A matrix you can solve for every right-hand side. It can be undone, so every answer b can be reached.
singular: A matrix that is not invertible. Its columns do not cover all of the space, so some right-hand sides cannot be reached.
linear combination: A mix of vectors made by multiplying each one by a number and then adding the results together.
dot product: A way to multiply a row of numbers by a list of numbers: multiply matching pairs and add them up.

Example

Think of mixing paint from a few base colors. If you only have red and yellow, you can only make colors that lie between them, so many colors are out of reach. If your base colors are spread out in different directions, you can mix almost any color you want. The lesson uses the same idea: columns lying flat in one plane cannot reach every target.

3 checked practice questions for this part are in the app.

Lecture 2: Elimination with matrices →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. MIT OpenCourseWare · CC BY-NC-SA 4.0 · not affiliated with or endorsed by MIT. These notes are shared under the same license.

LessonBound · Free lecture notes · Privacy · Terms · New York