7 Linear Maps

So far we have studied addition and multiplication by numbers of elements of vector spaces, and the structures which are generated by these two operations. Now we turn our attention to maps. In general a map \(T\) from a set \(V\) to a set \(W\) is a rule which assigns to each element of \(V\) an element of \(W\). For example, \(T(x,y):=(x^3y-4,\cos(xy))\) is a map from \(\mathbb{R}^2\) to \(\mathbb{R}^2\).

In Linear Algebra we focus on a special class of maps, namely linear maps – the ones which respect our fundamental operations, addition of vectors and multiplication by scalars. Some texts call these linear transformations, and in the case of \(V=W\) we may call this a linear operator.

Definition 7.1: (Linear map)

Let \(V\) and \(W\) be vector spaces over \(\mathbb{F}\). A map \(T :V\to W\) is called an \(\mathbb{F}\)-linear map if

  1. \(T(x+y)=T(x)+T(y)\), for all \(x, y\in V\),

  2. \(T(\lambda x)=\lambda T(x)\), for all \(\lambda\in\mathbb{F}\) and \(x\in V\).

If the field \(\mathbb{F}\) is clear from the context then we will often drop the \(\mathbb{F}\) and just refer to a linear map. Note that the addition and multiplication on the left hand sides of the expressions above are taking place in \(W\) and the addition and multiplication on the right hand sides are taking place in \(V\).

Let us note two immediate consequences of the definition.

Proposition 7.2:

Let \(V\) and \(W\) be vector spaces over \(\mathbb{F}\) and \(T:V \to W\) be a linear map, then

  1. \(T(\mathbf{0})=\mathbf{0}\),

  2. For arbitrary \(x_1, \cdots, x_k\in V\) and \(\lambda_1, \cdots,\lambda_k\in\mathbb{F}\) we have \[T(\lambda_1x_1+\cdots +\lambda_kx_k)=\lambda_1 T(x_1)+\cdots +\lambda_{k} T(x_k).\]

Proof.

  1. This follows from \(T(\lambda x)=\lambda T(x)\) if one sets \(\lambda=0\).

  2. This is obtained by applying the two conditions from the definition of a linear map repeatedly.

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Note that we can write the second property using the summation sign as well as \[T\bigg(\sum_{i=1}^k \lambda_ix_i\bigg) =\sum_{i=1}^k \lambda_iT(x_i) .\]

Linearity is a strong condition on a map. Let us consider some examples of what a linear map will look like.

Exercise 7.3:
Let \(V\) and \(W\) be vector spaces over \(\mathbb{F}\). Does there exist a map \(T:V \to W\) such that \(T(\mathbf{0} )=\mathbf{0}\) but \(T\) is not linear?

Let us now have a look at some examples of linear maps.

Example 7.4:
Let \(T:\mathbb{R}^3 \to \mathbb{R}^2\) be defined by \(T\begin{pmatrix}x_1\\x_2\\x_3 \end{pmatrix}=\begin{pmatrix} x_1+x_3 \\ 4x_2\end{pmatrix}\). Then for any \(x, y \in \mathbb{R}^3\) and \(\lambda \in \mathbb{R}\) we have that \[\begin{align*} T(x+y)&=T\begin{pmatrix}x_1+y_1\\x_2+y_2\\x_3+y_3 \end{pmatrix}=\begin{pmatrix}(x_1+y_1)+(x_3+y_3)\\4(x_2+y_2)\end{pmatrix}\\&=\begin{pmatrix} x_1+x_3 \\ 4x_2\end{pmatrix}+\begin{pmatrix} y_1+y_3 \\ 4y_2\end{pmatrix}=T(x)+T(y) \end{align*}\] and \[\begin{align*} T(\lambda x)&=T\begin{pmatrix}\lambda x_1\\\lambda x_2\\\lambda x_3 \end{pmatrix}=\begin{pmatrix} \lambda x_1+\lambda x_3 \\ 4\lambda x_2\end{pmatrix}\\ &=\lambda\begin{pmatrix} x_1+x_3 \\ 4x_2\end{pmatrix}=\lambda T(x). \end{align*}\] So \(T\) is an \(\mathbb{R}\)-linear map.
Example 7.5:
Let \(T:\mathbb{R}^2 \to \mathbb{C}\) be defined by \(T\begin{pmatrix}x_1\\x_2\end{pmatrix}=x_1 +i x_2\). Then for any \(x, y \in \mathbb{R}^2\) and \(\lambda \in \mathbb{R}\) we have that \[\begin{align*} T(x+y)&=T\begin{pmatrix}x_1+y_1\\x_2+y_2 \end{pmatrix}=(x_1+y_1)+i(x_2+y_2)\\&= (x_1+ix_2) + (y_1+iy_2)=T(x)+T(y) \end{align*}\] and \[\begin{align*} T(\lambda x)&=T\begin{pmatrix}\lambda x_1\\\lambda x_2 \end{pmatrix}=\lambda x_1+i\lambda x_2 \\&=\lambda (x_1+ix_2)\lambda T(x). \end{align*}\] So \(T\) is an \(\mathbb{R}\)-linear map. This formalises the idea we saw in Chapter 2 about how we can identify complex numbers with vectors in \(\mathbb{R}^2\).
Exercise 7.6:
Consider \(T:\mathbb{C}^2 \to \mathbb{C}^3\) defined by \(T\begin{pmatrix}x_1\\x_2\end{pmatrix}=\begin{pmatrix}(2+i)x_1\\i\\x_1+x_2 \end{pmatrix}\). Is \(T\) a \(\mathbb{C}\)-linear map?
Example 7.7:
The action of a matrix \(A\in M_n(\mathbb{R})\) is a linear map by Theorem 3.11.