In chapter 12 we discussed the linear transformations of ℝ^{2} and ℝ^{3} We now extend this concept for a general vector space.

**Definition 15.1.** *Let V and W be vector spaces over the same field F. A mapping*

T:V → W

*is called a linear transformation from V into W if it preserves vector addition and scalar multiplication, i.e.*

**Example 15.1.** *Let V(F) and W(F) be vector spaces. Then T*: *V* → *W defined by T{v) = Ow* ∀ u ∈ V *where Ow is the zero of W, is a linear transformation. It is called zero transformation as each element is mapped to zero. The transformation I*: *V → V defined by ...*

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