Fundamentals of Matrix Algebra, 2011a

Chapter 5
Graphical Exploraons of Vectors
2. Is T(k⃗x) = kT(⃗x)? Let’s arbitrarily pick k = 7, and use ⃗x as before.
([ ]) 21
T(7⃗x) = T
−14
[ ] −7
=
7
[ ] −1
= 7
1
= 7 · T(⃗x) !
So far it seems that T is indeed linear, for it worked in one example with arbitrarily
chosen vectors and scalar. Now we need to try to show it is always true.
Consider T(⃗x +⃗y). By the definion of T, we have
T(⃗x +⃗y) = A(⃗x +⃗y).
By Theorem 3, part 2 (on page 62) we state that the Distribuve Property holds for
matrix mulplicaon. 13 So A(⃗x + ⃗y) = A⃗x + A⃗y. Recognize now that this last part is
just T(⃗x) + T(⃗y)! We repeat the above steps, all together:
T(⃗x +⃗y) = A(⃗x +⃗y)
(by the definion of T in this example)
= A⃗x + A⃗y (by the Distribuve Property)
= T(⃗x) + T(⃗y) (again, by the definion of T)
Therefore, no maer what ⃗x and ⃗y are chosen, T(⃗x + ⃗y) = T(⃗x) + T(⃗y). Thus the first
part of the lineararity definion is sasfied.
The second part is sasfied in a similar fashion. Let k be a scalar, and consider:
T(k⃗x) = A(k⃗x)
(by the definion of T is this example)
= kA⃗x (by Theorem 3 part 3)
= kT(⃗x) (again, by the definion of T)
Since T sasfies both parts of the definion, we conclude that T is a linear transformaon.
.
We have seen two examples of transformaons so far, one which was not linear and
one that was. One might wonder “Why is linearity important?”, which we’ll address
shortly.
First, consider how we proved the transformaon in Example 99 was linear. We
defined T by matrix mulplicaon, that is, T(⃗x) = A⃗x. We proved T was linear using
properes of matrix mulplicaon – we never considered the specific values of A! That
is, we didn’t just choose a good matrix for T; any matrix A would have worked. This
13 Recall that a vector is just a special type of matrix, so this theorem applies to matrix–vector mulplica-
on as well.
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