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Linear Algebra: Companion notes

Abhay Soman

Section 4.1 Definition of linear transformation

We begin with an example. Consider the map \(T\colon\R^2\to\R^2\) defined by \(T\big((x,y)\big)=(-y,x)\text{.}\) This is the anticlockwise rotation by \(90^0\) in \(\R^2\text{.}\) This map has following two properties.
  1. For any \(v\in\R^2\) and any \(\alpha\in\R\) we have \(T(\alpha v)=\alpha T(v)\text{.}\)
  2. For any \(v,w\in\R^2\) we have \(T(v+w)=T(v)+T(w)\text{.}\)
The above two properties will be a part of the defining properties of linear transformations.
Let \(A=\begin{pmatrix}0\amp -1\\1\amp 0\end{pmatrix}\text{,}\) and define a map \(T_A\colon\R^2\to\R^2\) by
\begin{equation*} \begin{pmatrix}x\\y\end{pmatrix}\mapsto A\begin{pmatrix}x\\y\end{pmatrix}. \end{equation*}
We see that \(T_A\) satisfies the above two properties as well. Moreover, \(T\) and \(T_A\) define the same map.

Definition 4.1.1. (Linear Transformation).

Let \(V\) and \(W\) be vector spaces over a field \(F\text{.}\) A mapping \(T\colon V\to W\) is said to be a linear transformation over \(F\) or an \(F\)-linear transformation if for every \(\alpha\in F\) and any \(v,v_1,v_2\in V\) we have
  1. \(\displaystyle T(\alpha v)=\alpha T(v)\)
  2. \(T(v_1+v_2)=T(v_1)+T(v_2)\text{.}\)

Remark 4.1.2.

A linear transformation over \(F\) (resp., \(F\)-linear transformation) is also called a linear map over \(F\) (resp., an \(F\)-linear map).
  1. Show that any \(F\)-linear transformation maps the zero vector to the zero vector.
  2. Let \(T\colon V\to W\) be an \(F\)-linear transformation. Show that for every \(v\in V\) and any \(\alpha_i\in F\)
    \begin{equation*} T(\sum\alpha_iv_i)=\sum\alpha_iT(v_i)\text{.} \end{equation*}
  3. Show that \(T(v_1-v_2)=T(v_1)-T(v_2)\text{.}\)
  4. Show that a composition of \(F\)-linear transformation is an \(F\)-linear transformation.

Remark 4.1.4.

The first condition in Definition 4.1.1 may be considered as the commutativity between scalars and the map \(T\) (this is elaborated in Example 4.2.1). The second condition in Definition 4.1.1 corresponds to a group homomorphism between groups \((V,+)\) and \((W,+)\text{.}\)

Remark 4.1.5. (Linear extension of a map).

A linear transformation on a vector space is completely determined by its action on a basis. Indeed, if \(\{v_i\}\) is a basis of \(V\) and \(v\in V\) then \(v=\sum\alpha_iv_i\) for some \(\alpha_i\in F\text{.}\) Using properties of linear transformation we have \(T(v)=\sum\alpha_iT(v_i)\text{.}\) This process is called extending map linearly on \(V\) or linear extension of the map.

Definition 4.1.6.

The set of all \(F\)-linear mappings between vector spaces \(V\) and \(W\) over \(F\) is denoted by \(\Hom_F(V,W)\) or when the context is clear by \(\Hom(V,W)\text{.}\) We denote \(\Hom_F(V,V)\) by \(\End_F(V)\text{.}\) The set \(\Hom_F(V,F)\) is called the dual space of \(V\text{,}\) and it is denoted by \(V^*\text{.}\) The dual space \(V^*\) is in fact a vector space over \(F\) (see Remark 4.1.7). An element of \(V^*\) is called a linear functional.

Remark 4.1.7.

Note that \(\Hom_F(V,W)\) is a vector space over \(F\text{.}\) The addition is given by \((T+S)(v)=T(v)+S(v)\text{.}\) The scalar multiplication is given by
\begin{equation*} (\alpha T)(v)=\alpha\cdot T(v). \end{equation*}

Remark 4.1.8.

The vector space \(\End_F(V)\) is also a ring with unity if we take composition of functions as multiplicative operation and identity map as the unity.
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