Linear Algebra Linear Transformations-3, guaranteed and verified 100% Pass

1


Linear Transformations


Def. Let 𝑉 and 𝑊 be vector spaces. We call a function 𝑇: 𝑉 → 𝑊 a linear
transformation from 𝑉 to 𝑊 if for all 𝑢, 𝑣 ∈ 𝑉 and 𝑐 ∈ ℝ
a. 𝑇 (𝑢 + 𝑣 ) = 𝑇(𝑢) + 𝑇(𝑣)
b. 𝑇(𝑐𝑣 ) = 𝑐𝑇(𝑣).


Theorem: Let 𝑇: 𝑉 → 𝑊 be a linear transformation from a vector space 𝑉 to a
vector space 𝑊. Then for 𝑢, 𝑣, 𝑢1 , … 𝑢𝑛 ∈ 𝑉, 𝑎, 𝑏, 𝑎1 , … , 𝑎𝑛 ∈ ℝ
1. 𝑇(0) = 0
2. 𝑇(𝑣 − 𝑢) = 𝑇(𝑣 ) − 𝑇(𝑢)
3. 𝑇(𝑎𝑢 + 𝑏𝑣 ) = 𝑎𝑇(𝑢) + 𝑏𝑇(𝑣)
4. 𝑇(∑𝑛𝑖=1 𝑎𝑖 𝑢𝑖 ) = ∑𝑛𝑖=1 𝑎𝑖 𝑇(𝑢𝑖 )

Proof:

1. 𝑇(0) = 𝑇(2(0)) = 2𝑇(0) ⟹ 𝑇(0) = 0.
2. 𝑇(𝑣 − 𝑢) = 𝑇(𝑣 + (−𝑢)) = 𝑇(𝑣 ) + 𝑇(−𝑢)

= 𝑇(𝑣 ) − 𝑇(𝑢).
3. 𝑇(𝑎𝑢 + 𝑏𝑣 ) = 𝑇(𝑎𝑢) + 𝑇(𝑏𝑣 ) = 𝑎𝑇(𝑢) + 𝑏𝑇 (𝑣 ).
4. 𝑇(∑𝑛𝑖=1 𝑎𝑖 𝑢𝑖 ) = 𝑇(𝑎1 𝑢1 + 𝑎2 𝑢2 + ⋯ 𝑎𝑛 𝑣𝑛 )
= 𝑇(𝑎1 𝑢1 ) + 𝑇(𝑎2 𝑢2 + ⋯ + 𝑎𝑛 𝑢𝑛 )
= 𝑎1 𝑇(𝑢1 ) + 𝑇(𝑎2 𝑢2 ) + 𝑇(𝑎3 𝑢3 + ⋯ 𝑎𝑛 𝑢𝑛 )
⋮
= 𝑎1 𝑇(𝑢1 ) + 𝑎2 𝑇(𝑢2 ) + ⋯ + 𝑎𝑛 𝑇(𝑢𝑛 )
= ∑𝑛𝑖=1 𝑎𝑖 𝑇(𝑢𝑖 ).

, 2


Ex. Show that 𝑇: 𝑉 → 𝑊 is a linear transformation if and only if
𝑇(𝑐𝑣 + 𝑢) = 𝑐𝑇(𝑣 ) + 𝑇(𝑢) for all 𝑢, 𝑣 ∈ 𝑉, 𝑐 ∈ ℝ.


Proof: Case #3 of the previous theorem shows if 𝑇 is linear then
𝑇(𝑐𝑣 + 𝑢) = 𝑐𝑇(𝑣 ) + 𝑇(𝑢) for all 𝑢, 𝑣 ∈ 𝑉, 𝑐 ∈ ℝ.


Now let’s show that if 𝑇(𝑐𝑣 + 𝑢) = 𝑐𝑇 (𝑣 ) + 𝑇(𝑢) for all 𝑢, 𝑣 ∈ 𝑉, 𝑐 ∈ ℝ
then 𝑇 is linear.
We must show:
1. 𝑇(𝑢 + 𝑣 ) = 𝑇(𝑢) + 𝑇(𝑣) for all 𝑢, 𝑣 ∈ 𝑉
2. 𝑇(𝑐𝑣 ) = 𝑐𝑇(𝑣) for all 𝑐 ∈ ℝ.


1. Since 𝑇(𝑐𝑣 + 𝑢) = 𝑐𝑇 (𝑣 ) + 𝑇(𝑢) for all 𝑢, 𝑣 ∈ 𝑉, 𝑐 ∈ ℝ, it’s true
for 𝑐 = 1.
Thus 𝑇(𝑣 + 𝑢) = 𝑇(𝑣 ) + 𝑇(𝑢) for all 𝑢, 𝑣 ∈ 𝑉.


2. If we take 𝑢 = 0 then 𝑇(𝑐𝑣 + 0) = 𝑐𝑇(𝑣 ) + 𝑇(0)
⟹ 𝑇(𝑐𝑣 ) = 𝑐𝑇(𝑣 ) + 0 = 𝑐𝑇(𝑣).

, 3


Ex. Show that 𝑇: ℝ2 → ℝ2 by 𝑇 (< 𝑎1 , 𝑎2 >) =< 𝑎1 + 2𝑎2 , 𝑎1 > is a linear
transformation.


By the previous example we just need to show that 𝑇(𝑐𝑣 + 𝑢) = 𝑐𝑇(𝑣 ) + 𝑇(𝑢)
for all 𝑢, 𝑣 ∈ ℝ2 , 𝑐 ∈ ℝ.
For any 𝑢, 𝑣 ∈ ℝ2 , we have 𝑢 =< 𝑥1 , 𝑦1 > , 𝑣 =< 𝑥2 , 𝑦2 > and
𝑇(𝑐 < 𝑥1 , 𝑦1 > +< 𝑥2 , 𝑦2 >) = 𝑇(< 𝑐𝑥1 + 𝑥2 , 𝑐𝑦1 + 𝑦2 >)
=< 𝑐𝑥1 + 𝑥2 + 2(𝑐𝑦1 + 𝑦2 ), 𝑐𝑥1 + 𝑥2 >.


𝑐𝑇(< 𝑥1 , 𝑦1 >) + 𝑇(< 𝑥2 , 𝑦2 >) = 𝑐 (< 𝑥1 + 2𝑦1 , 𝑥1 >) + (< 𝑥2 + 2𝑦2 , 𝑥2 >)
=< 𝑐𝑥1 + 2𝑐𝑦1 + 𝑥2 + 2𝑦2 , 𝑐𝑥1 + 𝑥2 >
=< 𝑐𝑥1 + 𝑥2 + 2(𝑐𝑦1 + 𝑦2 ), 𝑐𝑥1 + 𝑥2 >
= 𝑇(𝑐 < 𝑥1 , 𝑦1 > +< 𝑥2 , 𝑦2 >).


So 𝑇(𝑐𝑢 + 𝑣 ) = 𝑐𝑇 (𝑢) + 𝑇(𝑣) for all 𝑢, 𝑣 ∈ ℝ2 , 𝑐 ∈ ℝ and 𝑇 is linear.