11.2MH1 LINEAR ALGEBRA
EXAMPLES 7: LINEAR TRANSFORMATIONS
1. Which of the following are linear transformations ?
(a) T1 :
2
2
such that T1
x
y
x
x
(b) T2 :
2
2
such that T2
x
y
x
x
(c) T3 :
2
2
such that T3
x
y
x y
sin y
(d) T4 : M22
M22 such that T4 A
At
(e) T5 : M22
such that T5 A
det A.
y
y
2. Let U and V be vector spaces and let T : U
V be a linear transformation. Which of the
following statements are true and which are false ? Prove the statements that are true and
provide counterexamples to those that are false.
(a) If u1
V.
un is linearly dependent in U then T u1
(b) If u1
in V .
un is linearly independent in U then T u1
(c) If u1
un span U then T u1
3. Let T :
2
2
1
0
(a) relative to the basis
4. Let D : Pn
x
y
such that T
T un
x
2x
y
y
0
1
T un
T un
is linearly independent
spans V .
. Find the matrix which represents T
; (b) relative to the basis
1
2
1
1
2
a1
Pn be the differentiation operator defined by
D an xn
an 1 xn
1
a1 x
a0
nan xn
1
n
Find the matrix which represents D relative to the basis 1 x
x
x y z
y
x 3y z
z
3x y 2z
Find bases for the range space and kernel of T .
5. Let T :
is linearly dependent in
3
3
such that T
1 an 1 xn
xn .
.
6. Let T : P2 P3 be the linear transformation defined by T p x
and kernel of T and hence determine the rank and nullity of T .
xp x . Find the range space
7. Give examples of linear transformations T as specified below.
(a) T :
5
3
has rank 2;
(b) T : M22
M22 has rank 3.
In each case find the nullity of T and give bases for the null spaces and range spaces of T .
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