4 Linear algebra

4.13 Linear maps

4.13.1 Motivation

Suppose that you have two finite sets X and Y and a function f:XY. If you know that f is onto then you get some information about X and Y: you know that X must be at least as large as Y.

But an arbitrary function between two vector spaces doesn’t necessarily give you any information about their relationship as vector spaces. To get such information, we need to restrict to functions that respect the vector space structure — that is, the scalar multiplication and the vector addition.

Functions with this property, which we’re going to define shortly, are called linear maps. They allow us to do something similar to the finite set example above: for example, if you have a surjective linear map from a vector space X to another vector space Y, it is true that dimXdimY.

4.13.2 Definition of a linear map

Definition 4.13.1.

Let V and W be vector spaces over the same field 𝔽. A function T:VW is called a linear map or a linear transformation if

  1. 1.

    for all λ𝔽 and all 𝐯V we have T(λ𝐯)=λT(𝐯), and

  2. 2.

    for all 𝐯,𝐰V we have T(𝐯+𝐰)=T(𝐯)+T(𝐰).

Point 1 is what it means to say that T respects scalar multiplication and point 2 is what it means to say that T respects vector addition.

This concept is so common that it has many names. For us,

  • T is a linear map

  • T is a linear function

  • T is a linear transformation

  • T is linear

all mean exactly the same thing, namely that T satisfies Definition 4.13.1.

4.13.3 Examples of linear maps

  1. 1.

    For any vector space V, the identity map id:VV and the zero map z:VV given by z(v)=𝟎V for all vV are linear.

  2. 2.

    Let A be a m×n matrix with entries in a field 𝔽. Then TA:𝔽n𝔽m defined by TA(𝐱)=A𝐱 is linear.

  3. 3.

    T:Mn×n()Mn×n(),T(A)=A2 is not linear.

  4. 4.

    T:n,T(x1xn)=i=1nxi is linear.

  5. 5.

    D:n[x]n[x] given by D(f)=dfdx is linear.

Let’s look at why some of these are true, starting with example 2. To show that TA is a linear map we have to check the two parts of the definition of being a linear map. Both of these are going to follow from properties of matrix multiplication and addition that you learned in the previous section.

  1. 1.

    Let 𝐱𝔽n and λ𝔽. Then

    TA(λ𝐱) =A(λ𝐱) definition of TA
    =λA𝐱 matrix multiplication properties
    =λTA(x) definition of TA
  2. 2.

    Let 𝐱,𝐲𝔽n. Then

    TA(𝐱+𝐲) =A(𝐱+𝐲) definition of TA
    =A𝐱+A𝐲 matrix multiplication properties
    =TA(𝐱)+TA(𝐲) definition of TA

The properties of matrix multiplication used were proved in Proposition 3.4.1.

Similarly, the fact that the differentiation map D of example 5 is linear follows from standard properties of derivatives: you know, for example, that for any two functions (not just polynomials) f and g we have ddx(f+g)=dfdx+dgdx, which shows that D satisfies the second part of the linearity definition.

As an example where the linearity of a map doesn’t just come from standard facts you already know, consider

T:2T(xy)=2xy

To show T is linear we have to show that it has properties 1 and 2 from the definition.

  1. 1.
    T(λ(xy)) =T(λxλy)
    =2λxλy
    =λ(2xy)
    =λT(xy)
  2. 2.
    T((x1y1)+(x2y2)) =T(x1+x2y1+y2)
    =2(x1+x2)(y1+y2)
    =(2x1y1)+(2x2y2)
    =T(x1y1)+T(x2y2).

Here are some examples of things which are not linear maps:

Example 4.13.1.
  • T:,T(x)=|x| isn’t linear. It doesn’t satisfy either linearity property. T(23)2T(3), and T(1+1)T(1)+T(1).

  • T:33,T(𝐱)=𝐱+(100). Again it doesn’t satisfy either part of the definition - you should check that.