Fundamentals of Matrix Algebra, 2011a

5.2 Properes of Linear Transformaons
and from Example 100
⎡
⎤
⎛⎡
T 100
⎝⎣ x ⎤⎞
x 1 + x 2
1
x 2
⎦⎠ = ⎢ 3x 1 − x 3
⎥
⎣ 2x
x 2 + 5x 3
⎦ ,
3
4x 1 + 3x 2 + 2x 3
where we use the subscripts for T to remind us which example they came from.
We found that T 98 was not a linear transformaon, but stated that T 100 was (although
we didn’t prove this). What made the difference?
Look at the entries of T 98 (⃗x) and T 100 (⃗x). T 98 contains entries where a variable is
squared and where 2 variables are mulplied together – these prevent T 98 from being
linear. On the other hand, the entries of T 100 are all of the form a 1 x 1 + · · · + a n x n ; that
is, they are just sums of the variables mulplied by coefficients. T is linear if and only if
the entries of T(⃗x) are of this form. (Hence linear transformaons are related to linear
equaons, as defined in Secon 1.1.) This idea is important.
.
Condions on Linear Transformaons
. Key Idea 16
Let T : R n → R m be a transformaon and consider the
entries of
⎛⎡
⎤⎞
x .
1
x 2
T(⃗x) = T ⎜⎢
. ⎥⎟
⎝⎣
⎦⎠ .
x n
T is linear if and only if each entry of T(⃗x) is of the form a 1 x 1 +
a 2 x 2 + · · · a n x n .
Going back to our baseball example, the manager could have defined his transformaon
as
⎛⎡
⎤⎞
x 1 [ ]
T ⎜⎢
x 2
⎥⎟
⎝⎣
x 3
⎦⎠ = x1 + x 2 + x 3 + x 4
.
x 1 + 2x 2 + 3x 3 + 4x 4
x 4
Since that fits the model shown in Key Idea 16, the transformaon T is indeed linear
and hence we can find a matrix [ T ] that represents it.
Let’s pracce this concept further in an example.
. Example 102 .Using Key Idea 16, determine whether or not each of the following
transformaons is linear.
([ ]) [ ] ([ ]) [ ]
x1 x1 + 1
x1 x1 /x
T 1 =
T
x 2 x 2 = 2 √x2
2 x 2
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