Definitions, examples and properties

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:RnRm which preserves sums and scalar multiples. Specifically, for all vectors u and v in Rm and all scalars c:

T(u + v) = T(u) + T(v)

 

(The transform of a sum of two vectors in Rn is the sum of their transformed vectors in Rm.)

T(cu) = cT(u)

 

 

(The transform of a scalar multiple of a vector in Rn is that same scalar multiple of the transformed vector in Rm.)

 

Examples of linear transformation

The zero transformation O:RnRm which takes every vector in Rn into the zero vector in Rm is a linear transformation.

  • It preserves sums: for any vectors u and v in Rn,
             O(u) + O(v) = 0 + 0 = 0 = O(u + v).
  • It preserves scalar multiples: for any scalar c and any vector u in Rn,
             cO(u) = c0 = 0 = 0 = O(cu).

The identity transformation I:RnRn which maps every vector in Rn into itself is a linear transformation.

  • It preserves sums: for any vectors u and v in Rn,
             I(u + v) = u + v = I(u) + I(v) .
  • It preserves scalar multiples: for any vector u in Rn and any scalar c,
             I(cu) = cu = cI(u).

For any unit vector a in Rn, look at the projection onto a as a function, i.e. proja:Rn → Rn , where proja(u) = (au)a for all u in Rn. Then proja is a linear transformation.

  • It preserves sums: for any vectors u and v in Rn,

proja(u + v)

= (a•[u + v])a
= (au + av)a
= (au)a + (av)a
= proja(u) + proja(v).

  • It preserves scalar multiples: for any vector u in Rn and any scalar c,

proja(cu)

= (a•[cu])a
= c(au)a
= c proja(u).

 

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:RnRm is a linear transformation and u is any vector in Rn. Then

T must preserve zero vectors: T(0) = 0.

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.

T must preserve negatives: T(–u) = – T(u).

Proof: T(–u) = T([–1]u) = [–1]T(u) = –T(u).

T must preserve differences: T(uv) = T(u) – T(v).

Proof: T(uv) = 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:RnRm is a linear transformation, v1. v2, ..., vk are vectors in Rn and c1, c2, ..., ck are scalars, then

T(c1v1 + c2v2 + ... + ckvk) = c1T(v1) + c1T(v2) + ... + ckT(vk).