Zero, Identity, Inclusion, and Projection Maps

Maps Are Specified by Their Inputs and Outputs

Some of the simplest linear maps appear whenever vector spaces are studied: a map that sends every input to zero, a map that leaves every input unchanged, a map that regards a subspace vector as a vector in a larger space, and a map that selects one coordinate or component. These maps are useful not only because their formulas are simple. Each makes the role of a map’s domain and codomain explicit. A formula alone does not specify a map unless we also know what kinds of objects it accepts and what kinds of objects it returns.

Unless stated otherwise, let the vector spaces in this tutorial be over the same field \(\mathbb{F}\), which may be \(\mathbb{R}\) or \(\mathbb{C}\). A map \(T:V\to W\) is linear if, for every \(\mathbf{u},\mathbf{v}\in V\) and every scalar \(c\in\mathbb{F}\), it satisfies both additivity and homogeneity:

The zero vectors of \(V\) and \(W\) need not be the same object, so we write \(\mathbf{0}_V\) and \(\mathbf{0}_W\) when the distinction matters. Similarly, the domain tells us where inputs come from, while the codomain specifies the space in which outputs are declared to lie. In the examples below, the spaces are finite-dimensional familiar spaces, but the definitions and linearity proofs do not require finite dimension.

The Zero Map and the Identity Map

The zero map’s domain and codomain can be different spaces. Its output must be the zero vector in the codomain. The identity map, by contrast, has the same space as both domain and codomain: it returns each input as an element of that space.

Proof. For the zero map, take any \(\mathbf{u},\mathbf{v}\in V\) and \(c\in\mathbb{F}\). By its definition, \(Z(\mathbf{u}+\mathbf{v})=\mathbf{0}_W\), while \(Z(\mathbf{u})+Z(\mathbf{v})=\mathbf{0}_W+\mathbf{0}_W=\mathbf{0}_W\). Also, \(Z(c\mathbf{u})=\mathbf{0}_W=c\mathbf{0}_W=cZ(\mathbf{u})\). Thus both conditions for linearity hold. For the identity map, \(\operatorname{id}_V(\mathbf{u}+\mathbf{v})=\mathbf{u}+\mathbf{v}=\operatorname{id}_V(\mathbf{u})+\operatorname{id}_V(\mathbf{v})\), and \(\operatorname{id}_V(c\mathbf{u})=c\mathbf{u}=c\operatorname{id}_V(\mathbf{u})\). Therefore it is linear as well. \(\square\)

Inclusion of a Subspace

A subset \(U\) of a vector space \(V\) is a subspace if it is itself a vector space using the addition and scalar multiplication inherited from \(V\). In particular, if \(\mathbf{u},\mathbf{v}\in U\) and \(c\in\mathbb{F}\), then \(\mathbf{u}+\mathbf{v}\in U\) and \(c\mathbf{u}\in U\). These closure properties allow a map to take an element of \(U\) and regard it as an element of \(V\).

Proof. Take \(\mathbf{u},\mathbf{v}\in U\) and \(c\in\mathbb{F}\). Since \(U\) is a subspace, \(\mathbf{u}+\mathbf{v}\in U\) and \(c\mathbf{u}\in U\), so both sides below are defined. By the inclusion formula and the inherited operations, \(i(\mathbf{u}+\mathbf{v})=\mathbf{u}+\mathbf{v}=i(\mathbf{u})+i(\mathbf{v})\), and \(i(c\mathbf{u})=c\mathbf{u}=ci(\mathbf{u})\). Thus the map is additive and homogeneous. \(\square\)

Coordinate Projections

For vector spaces \(V_1,\dots,V_k\) over the same field, their product \(V_1\times\cdots\times V_k\) consists of ordered tuples \((\mathbf{v}_1,\dots,\mathbf{v}_k)\) with \(\mathbf{v}_j\in V_j\). Addition and scalar multiplication are performed component by component. For an index \(j\) with \(1\le j\le k\), a coordinate projection selects the \(j\)-th component.

Proof. Let \(\mathbf{x}=(\mathbf{v}_1,\dots,\mathbf{v}_k)\) and \(\mathbf{y}=(\mathbf{w}_1,\dots,\mathbf{w}_k)\) be in the product, and let \(c\in\mathbb{F}\). The \(j\)-th component of \(\mathbf{x}+\mathbf{y}\) is \(\mathbf{v}_j+\mathbf{w}_j\). Therefore \(\pi_j(\mathbf{x}+\mathbf{y})=\mathbf{v}_j+\mathbf{w}_j=\pi_j(\mathbf{x})+\pi_j(\mathbf{y})\). The \(j\)-th component of \(c\mathbf{x}\) is \(c\mathbf{v}_j\), so \(\pi_j(c\mathbf{x})=c\mathbf{v}_j=c\pi_j(\mathbf{x})\). Both linearity conditions hold. \(\square\)

Common Mistakes About These Maps

Confusing the zero vector with the zero map. The zero vector is an element of a vector space. The zero map is a function between two spaces that returns the codomain’s zero vector for every input. For instance, \(Z:\mathbb{R}^2\to\mathbb{R}^3\) is not itself the vector \((0,0,0)\); it is the rule \((x,y)\mapsto(0,0,0)\).

Leaving the codomain unchanged after projecting. A common incorrect type declaration is to write \(\pi_{2,4}:\mathbb{R}^4\to\mathbb{R}^4\) while defining \(\pi_{2,4}(x_1,x_2,x_3,x_4)=(x_2,x_4)\). The output has two entries, so it belongs to \(\mathbb{R}^2\), not \(\mathbb{R}^4\). The corrected declaration is \(\pi_{2,4}:\mathbb{R}^4\to\mathbb{R}^2\). The domain and codomain are part of the map’s specification, not optional labels.

Treating an inclusion as a change to the vector. For the subspace \(U=\{(x,y,0)\}\subseteq\mathbb{R}^3\), the inclusion sends \((2,5,0)\) to \((2,5,0)\), not to \((2,5)\). Its codomain is \(\mathbb{R}^3\). A map that discards the third coordinate would instead be a coordinate projection with codomain \(\mathbb{R}^2\).

These maps provide basic examples against which other maps can be compared. When checking a proposed map, first identify its domain and codomain, then confirm that the operations used in the additivity and homogeneity equations are defined in the correct spaces. A correct formula with a mismatched output space is not a correctly specified map.