Chapter.

4 Linear Transformations
4.1 Definition and Examples

1. Linear Transformations
2. Linear Operators on 𝑅2
3. Linear Transformations from 𝑅𝑛 to 𝑅𝑚
4. Linear Transformations from 𝑉 to 𝑊
5. The Image and Kernel

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 1. Linear Transformations
Background
Linear mappings from one vector space to another play an important role in mathematics. This
chapter provides an introduction to the theory of such mappings.

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 1. Linear Transformations
Definition

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 2. Linear Operators on 𝑅 2

If 𝐿 is a linear transformation mapping a vector space 𝑉 into a vector space 𝑊, then it must
satisfy the following properties
1. Scalar multiplication and applying the function commute
𝐿 𝛼𝐯 = 𝛼𝐿(𝐯), (𝛼 = 𝛽 = 1) ... (a)
2. Vector addition and applying the function commute
𝐿 𝐯1 + 𝐯2 = 𝐿 𝐯1 + 𝐿(𝐯2 ), (𝐯 = 𝐯1 , 𝛽 = 0) ... (b)
Conversely, if 𝐿 satisfies properties (a) and (b), then
𝐿 𝛼𝐯1 + 𝛽𝐯2 = 𝐿 𝛼𝐯1 + 𝐿(𝛽𝐯2 )
= 𝛼𝐿 𝐯1 + 𝛽𝐿(𝐯2 )

Thus 𝐿 is a linear transformation if and only if 𝐿 satisfies the above properties (a) and (b).
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 2. Linear Operators on 𝑅 2

Notation:

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 2. Linear Operators on 𝑅 2

(a)

(b)

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 2. Linear Operators on 𝑅 2

Two steps combined

See examples.3 and 4 on page.187, 188 7

 𝑛 𝑚
3. Linear Transformations from 𝑅 to 𝑅
Example:
𝑥1 𝑦1
Suppose we have two vectors 𝐱 = 𝑥 , 𝐲 = 𝑦 , which is in 𝑅2 we want to transform these
2 2
3 2
vectors into 𝑅 , 𝐿: 𝑅 → 𝑅 , 3
𝑥2
Let is our linear transformation 𝐿 𝐱 = 𝑥1
𝑥1 + 𝑥2

𝑥1 α𝑥1 𝑥1 𝑦1 𝑥1 + 𝑦1
α𝐱 = α 𝑥 = α𝑥 , 𝐱+𝐲= 𝑥 + 𝑦 = 𝑥 +𝑦
2 2 2 2 2 2

α𝑥2 α𝑥2 𝑥2
(a) L αx = α𝑥1 = α𝑥1 =α 𝑥1 = α𝐿(𝐱)
α𝑥1 + α𝑥2 α(𝑥1 + 𝑥2 ) 𝑥1 + 𝑥2 8
 𝑛 𝑚
3. Linear Transformations from 𝑅 to 𝑅
Example: (continued)
𝑥2 + 𝑦2 𝑥2 + 𝑦2 𝑥2 + 𝑦2
(b) L 𝐱 + 𝐲 = 𝑥1 + 𝑦1 = 𝑥1 + 𝑦1 = 𝑥1 + 𝑦1
𝑥1 + 𝑦1 + 𝑥2 + 𝑦2 𝑥1 + 𝑥2 +𝑦1 +𝑦2 (𝑥1 +𝑥2 ) +(𝑦1 +𝑦2 )

𝑥2 𝑦2
L 𝐱+𝐲 = 𝑥1 + 𝑦1 = 𝐿 𝐱 + 𝐿(𝐲)
𝑥1 + 𝑥2 𝑦1 + 𝑦2

𝑥2
Therefore 𝐿 𝐱 = 𝑥1 doesn’t infract linear transformation, i.e. 𝐿 𝐱 satisfied the
𝑥1 + 𝑥2
properties of the linear transformation.
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 𝑛 𝑚
3. Linear Transformations from 𝑅 to 𝑅
Example:
𝑥1 1 0
Suppose we have two vectors 𝑥 = , which is in 𝑅2 we want to transform these vectors to 𝑅3 ,
2 0 1
𝐿: 𝑅2 → 𝑅 3 ,
𝑥2
Let is our linear transformation 𝐿 𝐱 = 𝑥1
𝑥1 + 𝑥2
𝑥2 0 𝑥2 1
𝑥1 1 𝑥1 0
𝑥1 𝑥1
𝑥2 = 0 : 𝐿 𝐱 = = 1 and 𝑥 =
2 1
: 𝐿 𝐱 = = 0
𝑥1 + 𝑥2 1 𝑥1 + 𝑥2 1
0 1
∴ 𝐴 = 1 0 which is in 𝑅3 .
1 1

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 𝑛 𝑚
3. Linear Transformations from 𝑅 to 𝑅

(Book)

See example 5,6 in book, page.189 11

4. Linear Transformations from 𝑉 to 𝑊

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 4. Linear Transformations from 𝑉 to 𝑊

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See example.10
5. The Image and Kernel

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 5. The Image and Kernel
It is obvious that ker(L) is nonempty since 0V, the zero vector of V, is in ker(L).

Proof (i): It is obvious that ker(𝐿) is nonempty since 𝟎v , the zero vector of V, is in ker(𝐿).
(a). Scalar multiplication: 𝐯 ∈ ker(𝐿) and 𝛼 be a scalar then
𝐿 (𝛼𝐯) = 𝛼𝐿 (𝐯) = 𝛼𝟎w = 𝟎w ∈ ker(𝐿)
(b). Vector addition: let 𝐯1 , 𝐯2 ∈ ker(𝐿). Then
𝐿 (𝐯1 + 𝐯2 ) = 𝐿 (𝐯1 ) + 𝐿 (𝐯2 ) = 𝐨w + 𝐨w = 𝐨w ∈ ker(𝐿)
∴ ker(𝐿) is a subspace of 𝑉.

Proof (ii):
(a). Scalar multiplication:
Do by yourself
(b). Vector addition:

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 5. The Image and Kernel
Example 12:
a+(-a)=0
Let 𝐿: ℝ3 → ℝ2 be the linear transformation defined by 𝐿 𝐱 = (𝐱1 + 𝐱 2 , 𝐱 2 +𝐱 3 )T (-a)+(a)=0
∴ ker ℝ3 = (0,0)T
Let 𝑆 be the subspace of ℝ3 spanned by 𝐞1 and 𝐞3 , if 𝐱 ∈ ker(𝐿), then
𝐱1 + 𝐱 2 = 0 and 𝐱 2 + 𝐱 3 = 0, setting the free variable 𝐱 3 = 𝑎, we get 𝐱1 = 𝑎, 𝐱 2 = −𝑎
and hence ker(𝐿) is the one-dimensional subspace of ℝ3 consisting of all vectors of the form a(1, −1, 1) T .

If 𝐱 ∈ 𝑆, the 𝐱 must be of the form (a, 0, a)T , and hence 𝐿(𝐱) = (a, a)T . Clearly, 𝐿 𝑆 = ℝ2 . Since the image of the
subspace 𝑆 is all of ℝ2 , it follows that the entire range of 𝐿 must be ℝ2 [i.e., L ℝ3 = ℝ2 ].

a+(0)=a
(0)+(a)=a
∴ 𝑖𝑚𝑎𝑔𝑒 𝑜𝑓 𝑆, 𝐿(𝐱) = (𝑎, 𝑎)T
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