Page 178 - A Course in Linear Algebra with Applications

162 Chapter Six: Linear Transformationns

Theorem 6.2.1
Let T : V —> W be a linear transformation. Then

r(Ov) = o w

and

T(c 1 v 1 +c 2 v 2 + - • •+c fcv A!) = c 1 T(v 1 )+c 2 T(v 2 ) + - • -+c kT(v k)

/or a// vectors v$ and scalars Ci.
Thus a linear transformation always sends a zero vector
to a zero vector; it also sends a linear combination of vectors
to the corresponding linear combination of the images of the
vectors.
Proof
In the first place we have

T(0 V) = T(0 V + 0 V) = T(0 V) + T(0 V)

by the first defining property of linear transformations. Addi-
tion of — T(Oy) to both sides gives Ow = T(Oy), as required.
Next, use of both parts of the definition shows that

T(civi H h c fc_iv fc_! + c kv k)

is equal to the vector

r(cxvi H h Cfc_iVfc_i) + c fc r(v fc ).

By repeated application of this procedure, or more properly
induction on k, we obtain the second result.

Representing linear transformations by matrices
We now specialize the discussion to linear transformations
of the type