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M02.1 · Vectors as arrows and as data

A free, illustrated module of 4 lessons and one interactive in M02 Linear Algebra I: Vectors and Matrices: Vectors as arrows; Vectors as lists of numbers; Adding and scaling vectors; Data as vectors.

M02.1 L1 · Core 4 lessons · about 100 min Live · in review

Part of M02 Linear Algebra I: Vectors and Matrices in the Mathematics track.

A vector is two things at once: an arrow with a length and a direction, and a list of numbers that a computer can store. These four lessons build both pictures and the dictionary between them, introduce adding and scaling, the two operations on which all of linear algebra is built, with the eight rules they obey, and then show how houses, pictures and words become vectors whose distances can be measured.

What you need first. School algebra and coordinates in the plane, including Pythagoras’ theorem (M01.1 and M01.2). Sine and cosine (M01.5) appear only in one optional note.

Lessons

1. Vectors as arrows M02.1.1 25 min

   A vector is a length and a direction, drawn as an arrow that may sit anywhere. Learn to tell when two arrows are the same vector, to find an arrow’s length from its run and rise, and to meet the zero vector, before any coordinates are attached.

2. Vectors as lists of numbers M02.1.2 25 min

   Put an arrow’s tail at the origin and the coordinates of its tip pin it down. Learn to move between arrows and lists of numbers, to work with vectors in three and more dimensions where no picture exists, and to read a row of data as a vector.

3. Adding and scaling vectors M02.1.3 25 min

   Vectors can be added tip to tail and stretched by a number. See both operations as pictures and as arithmetic on coordinates, derive the eight rules they obey, and read a subtraction as the arrow that joins two tips.

4. Data as vectors M02.1.4 25 min

   A house, a picture and a word can each be written as a list of numbers, which makes each one a point in space. Learn to measure the distance between data points, see why the unit of a feature changes the answer, and take a first look at high dimensions.

The interactive

M02.1-X1 · Vector playground. Drag two arrows and watch their sum form tip to tail, with a scaled copy and the difference alongside. Every coordinate and length updates as a number while you move. It lives in Adding and scaling vectors.