Lessons · Maths · beginner · MIT OpenCourseWare

1. The Geometry of Linear Equations

The video “1. The Geometry of Linear Equations” by MIT OpenCourseWare as 6 lessons, with the idea explained, the code from the screen, steps to try in the practice terminal, practice problems with solutions and a quiz that leads back to the moment.

6lessons
5pieces of code
2formulas
11exercises
40min of video
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1. The Geometry of Linear Equations, video by MIT OpenCourseWare▶ 39:48 · MIT OpenCourseWare
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0:006 lessons across the whole video39:48

You will be able to

  • Explain why a system of linear equations can be read as lines or planes intersecting, using the row picture.
  • Rewrite a system of equations as a matrix equation Ax = b.
  • Describe the column picture of Ax = b as a linear combination of the columns of A.
  • Determine whether a system has one solution, no solution, or infinitely many solutions from its geometry.
  • Relate the row and column pictures to each other for 2×2 and 3×3 systems.

Good to know first

  • Solving two linear equations in two unknowns by substitution or elimination
  • Basic coordinate geometry: plotting points and graphing lines in the plane
  • Basic idea of a vector as an ordered list of numbers

Syllabus

6 lessons and a final project

  1. 1

    Course overview and the goal of linear algebra Free

    0:00–3:003 min of video

    This lesson introduces linear algebra as the study of systems of linear equations with n equations and n unknowns, seen through the row picture, the column picture, and matrix form. It shows why the course is built around these three views of the same problem.

    2 exercises3 questions
  2. 2

    The row picture: intersecting lines in 2D Beyond Pass

    3:00–10:207 min of video

    Two equations with two unknowns describe two straight lines, and the solution is the point where they cross. We check this by drawing each equation as a line and finding the intersection.

    1 code4 build steps2 exercises3 questions
  3. 3

    Combining columns: the column picture of Ax = b Beyond Pass

    10:20–16:206 min of video

    This lesson shows how the equation Ax = b can be read as asking which mix of the columns of A lands exactly on the vector b. You will see how scaling two columns and adding them gives a "linear combination" that reaches b.

    1 formula2 exercises2 questions
  4. 4

    The column picture: combining the columns of A Beyond Pass

    16:20–23:407 min of video

    The column picture reads the same system Ax = b as a single vector equation: the unknowns are scalars that multiply the columns of A, and we add them up to hit the target vector b. Solving the system means finding the right mix of columns that lands exactly on b.

    1 code1 formula3 build steps2 exercises3 questions
  5. 5

    Matrix form Ax = b and reading the same system two ways Beyond Pass

    23:40–30:006 min of video

    You learn to write a system of linear equations as one matrix equation Ax = b, and to read it two ways: row by row (each equation is a plane) and column by column (x, y, z are the weights of a combination of the columns that must land on b). The lesson ends with the question that drives the rest of the course: can we solve Ax = b for every b?

    1 code1 build step2 exercises4 questions
  6. 6

    Singular cases, no-solution examples and the column picture Beyond Pass

    30:00–39:4810 min of video

    This lesson shows how a system Ax = b can have no solution or a unique solution depending on whether the columns of A combine to reach b. It then introduces the column picture, where Ax is a combination of the columns of A.

    2 code3 build steps1 exercise3 questions

Final project

One System, Three Views: A Geometry of Linear Equations Portfolio

Choose a 2×2 system and a 3×3 system. Solve each one by the row picture, the column picture and matrix form Ax = b, then compare the pictures. Extend the work to singular cases to show when a system has no solution or infinitely many. The aim is to show that the three views describe the same problem.

Its 7 tasks open with Beyond Pass.

Free preview · Lesson 1 in full

Course overview and the goal of linear algebra

This lesson introduces linear algebra as the study of systems of linear equations with n equations and n unknowns, seen through the row picture, the column picture, and matrix form. It shows why the course is built around these three views of the same problem.

0:00 to 3:00 in the video

Scratchpad

Draw and write next to the video.

The idea

Linear algebra is the study of linear equations. The course in this video (MIT 18.06, taught by Gilbert Strang) starts with the most basic kind of problem: a system of equations where every unknown appears only to the first power, with no squares, no products of unknowns, and no unknowns inside other functions. The board in the lecture states the general goal as n linear equations with n unknowns.

The lecture gives you three pictures of the same problem. The first is the row picture: each equation is a line (in two unknowns) or a plane (in three), and the answer is the place where all of them meet. The second is the column picture, which the lecturer marks with a star because it is the one he cares about most: the same equations are read as a way of combining column vectors to produce a target vector. The third is matrix form, which packs all the numbers into one compact statement, Ax = b.

Why does this matter? A small system like 2x - y = 0 and -x + 2y = 3 can be solved by substitution, but real problems have hundreds or thousands of equations. The three pictures give you a way to think about such systems geometrically and algebraically, and the same ideas (combinations, independence, solutions) carry over to every size.

Take the example on the board: 2x - y = 0 and -x + 2y = 3. In the row picture, the first equation is a line through the origin and the second is another line; they cross at one point, which is the solution. In the column picture, you ask how to combine the columns (2, -1) and (-1, 2) with weights x and y to reach the target (0, 3). The matrix form writes the same numbers as a 2 by 2 matrix times the unknown vector (x, y) equals (0, 3).

So the course goal is to understand all three views, to see that they describe one object, and to use this to solve n by n systems and to study what can go wrong when a solution is missing or not unique.

Key points

  • The course is about systems of linear equations: n equations with n unknowns, where every unknown appears only to the first power. 1:30
  • Row picture: each equation is a line (2 unknowns) or plane (3 unknowns), and the solution is where they all meet. 1:30
  • Column picture (the one the lecturer stars): the solution is the set of weights that combine the columns of the coefficient matrix to reach the right-hand side. 2:00
  • Matrix form packs the whole system into one statement, Ax = b, with A the matrix of coefficients, x the unknowns and b the right-hand side. 2:00
  • The example 2x - y = 0, -x + 2y = 3 can be read in all three ways, and each view gives the same answer. 2:50

Practice

Explain Solution checked

In your own words, explain the difference between the row picture and the column picture for the system 2x - y = 0 and -x + 2y = 3.

Solution
  1. Row picture: each equation is a line in the x-y plane; the two lines meet at the solution point.
  2. Column picture: the columns (2, -1) and (-1, 2) are combined with weights x and y to reach the target (0, 3).
  3. Both pictures describe the same solution, just looked at from two different angles.

The row picture asks where two lines meet; the column picture asks which weights combine the columns to give the right-hand side. Both give the same solution.

Calculate Solution checked

Count the unknowns and equations in a system with 3 equations and 3 unknowns, then write the matrix form Ax = b. For the number of unknowns in the 3 by 3 case, what is the value of n?

Hint

n is the number of unknowns, which equals the number of equations in the square case.

Solution
  1. The system has 3 equations and 3 unknowns, so n = 3.
  2. The matrix A is 3 by 3, x has 3 entries and b has 3 entries.
  3. So the number asked for is n = 3.

3

On your scratchpad

On your scratchpad, draw two lines for 2x - y = 0 and -x + 2y = 3 on the same axes, and mark the point where they cross. Then write the two columns (2, -1) and (-1, 2) next to it.

Quiz

  1. What does the lecture say the general goal of the course is?

    1. Solving quadratic equations with n unknowns
    2. n linear equations with n unknowns
    3. Finding the derivative of a matrix
    4. Drawing graphs of curves with no unknowns
    Show the answer

    B. n linear equations with n unknowns · The board reads 'n linear equations, n unknowns', the core problem of the course.

    1:30 watch the moment

  2. Which picture does the lecturer mark with a star as the one he cares about most?

    1. The row picture
    2. The column picture
    3. The matrix form
    4. The substitution method
    Show the answer

    B. The column picture · The board marks 'Column picture' with a star.

    2:00 watch the moment

  3. In the row picture of 2x - y = 0 and -x + 2y = 3, what is each equation?

    1. A point
    2. A line
    3. A circle
    4. A number
    Show the answer

    B. A line · In two unknowns, each linear equation is a straight line; the solution is where the lines meet.

    2:50 watch the moment

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Lessons 2–6 and the final project

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Made by Beyond from the video “1. The Geometry of Linear Equations” by MIT OpenCourseWare. The video plays in YouTube's own player. Watch it on YouTube · All lessons