Page 111 - 35Linear Algebra
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6
Linear Transformations
The main objects of study in any course in linear algebra are linear functions:
Definition A function L: V → W is linear if V and W are vector spaces
and
L(ru + sv) = rL(u) + sL(v)
for all u, v ∈ V and r, s ∈ R.
Reading homework: problem 1
Remark We will often refer to linear functions by names like “linear map”, “linear
operator” or “linear transformation”. In some contexts you will also see the name
“homomorphism” which generally is applied to functions from one kind of set to the
same kind of set while respecting any structures on the sets; linear maps are from
vector spaces to vector spaces that respect scalar multiplication and addition, the two
structures on vector spaces. It is common to denote a linear function by capital L
as a reminder of its linearity, but sometimes we will use just f, after all we are just
studying very special functions.
The definition above coincides with the two part description in Chapter 1;
the case r = 1, s = 1 describes additivity, while s = 0 describes homogeneity.
We are now ready to learn the powerful consequences of linearity.
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