Clear Concept Commons

Course 01

Vectors and Matrices

A first course in linear algebra. Vectors stay arrows, matrices stay maps, and both pictures are still in view when you reach Ax = b.

Lessons
Eight
Setting
The plane, then space
Preparation
High-school algebra

About this course

Vectors and matrices are the objects linear algebra is about. This course builds them from geometry. You draw arrows, add them, measure an angle with a single product, and then let a matrix move every arrow in the plane at once.

It is written for a first meeting with the subject, and for anyone who has used matrices as a recipe and wants the picture that makes the recipe necessary.

You will need coordinates, the graph of a line, and the ability to solve two linear equations in two unknowns. You do not need any prior matrix algebra.

You will be able to

  • Move between an arrow, an ordered pair, and a column without losing track of the object.
  • Add vectors and scale them, and say what changed in the picture.
  • Compute a dot product and read it as a length, a projection, or an angle.
  • Describe the span of a set of vectors, and tell when one of them is redundant.
  • Treat a matrix as a function that sends every vector to another vector.
  • Multiply matrices by composing those functions, and see why the order matters.
  • Decide whether Ax = b has one solution, infinitely many, or none.

Syllabus

Eight lessons. Each one opens on a figure.

  1. Arrows and lists

    A vector is an arrow with a length and a direction. Choosing an origin turns that arrow into an ordered pair, and the pair is still the arrow.

  2. Addition and scaling

    Adding vectors is the parallelogram law. Multiplying by a scalar stretches or reverses the arrow and keeps it on the same line.

  3. Length and the dot product

    The dot product of a vector with itself is the square of its length. The dot product of two different vectors records the angle between them.

  4. Projection

    Drop a perpendicular from one vector onto another and the shadow you cut off is the projection. Its length comes from the dot product.

  5. Combinations and span

    A linear combination is a sum of scaled vectors. The span is every point those combinations can reach. A vector already in the span of the others is redundant.

  6. A matrix is a map

    A matrix sends each vector to a new vector. Its columns are the landing places of the standard basis arrows, and those columns determine the whole map.

  7. Multiplying matrices

    Applying one matrix and then another is itself a matrix. Building that matrix is multiplication, and the two orders are different trips.

  8. Solving Ax = b

    The product Ax is a combination of the columns of A. The equation asks whether b is in that span, and a solution x names the combination.

How to study

Work one lesson at a time. Draw the figure before you write an equation. When a calculation and a picture disagree, debug the picture first: the algebra is a record of it, not a replacement.