Linear transformations are functions which keep intact the basic vector operations of addition and scalar multiplication.
A linear transformation from Rn to Rm is a function T:Rn → Rm which preserves sums and scalar multiples. Specifically, for all vectors u and v in Rm and all scalars c: |
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Examples of linear transformation
The zero transformation O:Rn → Rm which takes every vector in Rn into the zero vector in Rm is a linear transformation.
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The identity transformation I:Rn → Rn which maps every vector in Rn into itself is a linear transformation.
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For any unit vector a in Rn, look at the projection onto a as a function, i.e. proja:Rn → Rn , where proja(u) = (a•u)a for all u in Rn. Then proja is a linear transformation.
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In the definition above, we described linear transformations by saying that they "preserve sums and scalar multiples". Linear transformations also preserve other vector related quantities.
Properties preserved by linear transformations |
Suppose T:Rn → Rm is a linear transformation and u is any vector in Rn. Then |
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Proof: T(0) = T(0u) = 0T(u) = 0. |
Note that the zero vectors on each side of this equation are from different spaces. If we wanted to be a bit more precise, we could write T(0n) = 0m or something similar, but we usually don't, since no other interpretation of this equation makes sense. |
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Proof: T(–u) = T([–1]u) = [–1]T(u) = –T(u). |
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Proof: T(u – v) = T(u + [–v]) = T(u) + T(–v) = T(u) – T(v). |
Since linear combinations of vectors are just extended combinations of sums and scalar multiples, linear transformations also preserve linear combinations.
If T:Rn → Rm is a linear transformation, v1. v2, ..., vk are vectors in Rn and c1, c2, ..., ck are scalars, then |
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