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# Linear mappings.

Definition 6.1.1   Let V and W be two vector spaces. A mapping

is a linear mapping or a linear operator if it verifies the following properties:
1.
.
2.
.

Proposition 6.1.2   Let V and W be two vector spaces. A mapping

is a linear mapping if, and only if, it verifies the following property:

Proposition 6.1.3   Let U,V,W be three vector spaces with respective dimension m,n,p, in which bases are given. Let and be two linear mappings. Then is a linear mapping.

Remark 6.1.4

If is the matrix of and w.r.t. the given bases, then the matrix of is

where

Proposition 6.1.5   If U and V are two vector spaces, the set of all the linear mappings from U to V is a vector space.

Remark 6.1.6   Let be a linear application.
• If is injective, then .
• If is surjective, then .
• If is bijective, then .

Next: Kernel and Image of Up: Linear mappings. Previous: Linear mappings.
Noah Dana-Picard
2001-02-26