Page 117 - DMTH502_LINEAR_ALGEBRA

Unit 7: Algebra of Linear Transformation

Theorem 2: Let { , , ... } be a basis for K . Extend this basis to any basis { , , ... , , Notes
1 2 v(T) T 1 2 v(T) v(T)+1
... } for v .
n n
Then { T, ..., T} is a basis for R .
v(T) + 1 n T
Proof: Let { , , ... } be any basis for v . Any vector of R is of the form T for some V . Let
1 2 v n T n
n
T a 1 i ;
i 1
then

n n n
T a i i T a i i T ) a i i T ),
T
i 1 i 1 i v ( ) 1
since

v
T
T for i 1, 2, ... ( )
i
Hence ( T, ..., T) spans R . As the dimension of R is not known we have to prove linear
v(T)+1 n T T
independence of the above vectors. Suppose scalars b , not all zero, exist such that
i
n n
b ( T ) b T
i i i i
v ( ) 1 i v ( ) 1
T
T
n
Then b i i K ; but { , ... v(T) } spans K , so for suitable scalars c i
1
T
T
T
v ( ) 1
t
n v ( )
b i i c i i
T
v ( ) 1 i 1
This contradicts the linear independence of { , ... } , so the vectors { , ... } are linearly
1 n v(T)+1 n
independent and therefore form a basis of R .
T
Theorem 3: If T is a linear transformation from V to w, then
n
p (T) + v(T) = n.
Self Assessment
1. In the space of all polynomials p(x) of all degrees define mapping M and D by:
d
D p(x) = p(x), M p(x) = x p(x)
dx
Find
(i) DM – MD
2
2
(ii) M D + MD
2. Let v be the infinite dimensional space of all real polynomials. Let D and J be the Linear
Transformation defined by
d
D p(x) = p(x)
dx
x
J p(x) = p ( )dt
t
0
for p(x) v,

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