MATH218 — Elementary Linear Algebra with Applications

2.1 · Coordinate space

An element of coordinate space is a list of numbers. In $F^n$ you write that list with $n$ coordinates, and you must be able to work with it even when no picture exists.

The zero list is the list $0 = (0, \ldots, 0)$ of length $n$ whose every coordinate is the number $0$. The same symbol $0$ is used for that list and for the number zero. The equation around the symbol shows which one is meant.

Drawings in $\mathbb{R}^2$ are only a guide. A list such as $(\sqrt{3}, -5, 8, \pi, \tfrac{1}{2})$ is an element of $\mathbb{R}^5$ with the same formal status as a point of $\mathbb{R}^2$, even though you cannot sketch it.

The zero list

The zero list in $F^n$ is $0 = (0, \ldots, 0)$: one list of length $n$, and every coordinate is the number $0$.

[!KEY] $0 = (0, \ldots, 0)$ is a list. Each coordinate $0$ inside it is the number zero.

You read the symbol from the equation that contains it. On the left of $0 = (0, \ldots, 0)$, the symbol $0$ names the whole list. On the right, each $0$ is the number zero.

That definition does not depend on a sketch. Drawings in $\mathbb{R}^2$ are only a guide

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Matrices — Exercises

Exercise 1

The zero list of length $n$ is $0 = (0, \ldots, 0)$, and a drawing in the plane is only a guide for lists you cannot sketch. That idea belongs to ____.

Worked solution

The zero list in $F^n$ is the list of length $n$ whose every coordinate is $0$, and the same symbol $0$ is used for the list and the number . A drawing in $\mathbb{R}^2$ is only a guide; a five-coordinate list still has the same formal status as a point of $\mathbb{R}^2$ .

Exercise 2

A drawing in the plane is only a guide, while a list such as $(\sqrt{3}, -5, 8, \pi, \tfrac{1}{2})$ is still an element of $\mathbb{R}^{5}$. The packed label for that reminder is ____.

Worked solution

Drawings in $\mathbb{R}^{2}$ are only a guide

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