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Linear Maps

Linear Transformation (Map)

Let \(V,W\) be vector spaces over \(F\) and let \(T:V\to W\) be a function.

We say \(T\) is a linear transformation from \(V\) to \(W\) if for all \(v_1,v_2\in V,\alpha\in F\), we have

\[ \begin{cases} T(v_1+v_2)=T(v_1)+T(v_2)\\ T(\alpha v_1)=\alpha T(v_1) \end{cases} \]

Also call \(T\) a linear map, or \(T\) is linear. It is also called a homomorphism between vector spaces \(V\) and \(W\).

Theorem

Let \(V\) and \(W\) be vector spaces over \(F\). The set

\[ Hom(V,W):=\{T:V\to W\mid T\text{ is linear}\} \]

is a vector space over \(F\).

Range and kernel

Let \(T\) be a linear transformation from \(V\) to \(W\).

The range of \(T\), denoted as \(\mathsf{R}(T)\), is the set

\[ \mathsf{R}(T):=\{T(v)\mid v\in V\}. \]

The kernel of \(T\), denoted as \(\mathrm{ker}(T)\), is the set

\[ \mathrm{ker}(T):=\{v\in V\mid T(v)=0_W\}. \]
One step verification

Let \(V,W\) be vector spaces over \(F\) and let \(T:V\to W\) be a function.

\(T\) is linear if and only if for all \(v_1,v_2\in V,\alpha\in F\), we have \(T(v_1+\alpha v_2)=T(v_1)+\alpha T(v_2)\).

Properties of linear transformation

Let \(T\) be a linear transformation from vector spaces \(V\) to \(W\).

  1. \(T(0_V)=0_W\)
  2. \(\mathsf{R}(T)\) and \(\mathrm{ker}(T)\) are subspaces of \(W\) and \(V\), respectively.
  3. If \(U:V\to W\) is linear, then \(T+U\) is also linear.
  4. \(\alpha T\) is linear, for any \(\alpha\in F\).
  5. \(\ker{T}=\{0\}\) if and only if \(T\) is injective.
Basis preservation

Let \(T\) be an invertible linear transformation from vector spaces \(V\) to \(W\) and let \(S_1,S_2\) be bases for \(V,W\) respectively.

\(S_1\) is a basis for \(V\) if and only if \(S_2\) is a basis for \(W\).

Example

Let \(V=\mathsf{C}([a,b],\mathbb{R})\) and \(F=\mathbb{R}\) where \(a,b\in\mathbb{R}\). Let \(T:V\to\mathbb{R}\) be defined by

\[ T(f):=\int_{a}^{b}f(x)dx. \]

Then \(T\) is linear.