6.1 Definition and Examples
Definition 6.1.1. Let \(A,B\) be vector spaces over a field \(\mathbb {F}\). A linear transformation from \(A\) to \(B\) is a function \(T:A\longrightarrow B\) such that \[T(\alpha X+Y)=\alpha T(X)+T(Y)\] for all \(X,Y\in A\) and \(\alpha \in \mathbb {F}\).
The single condition above is equivalent to the two conditions \(T(X+Y)=T(X)+T(Y)\) and \(T(\alpha X)=\alpha T(X)\), which are often quoted separately: take \(\alpha =1\) for the first, and \(Y=0\) for the second.
Remark.
- i.
- Sometimes we write \(Tx\) for \(TX\).
- ii.
- The set of all linear transformations from \(A\) to \(B\) is denoted by \(\mathcal {L}(A,B)\).
- i.
- Zero map i.e. \(0\in \mathcal {L}(A,B)\) defined by \(0x=0\), \(\forall x\in A\).
- ii.
- Identity map i.e. \(I\in \mathcal {L}(A,B)\) defined by \(Ix=x\).
- iii.
- Differentiation i.e. \(T\in \mathcal {L}(P(\mathbb {R}),P(\mathcal {R}))\) defined by \(TP=P'\).
- iv.
- Integration i.e. \(\mathcal {L}(P(\mathbb {P}),\mathbb {R})\) defined by \(TP=\int \limits ^1_0P(x)dx\).
- v.
- Back shift operator, i.e. \(T\in \mathcal {L}(\mathbb {F}^{\infty },\mathbb {F}^{\infty })\) defined by \(T(x_1,x_2,x_3,\dots )=(x_2,x_3,x_4,\dots )\) — the first term is discarded and everything moves back one place.
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