LINEAR ALGEBRA(MA-102)
IIT ROORKEE
Assignment-3 Spring Sem:2016-17
1. Which of the following functions T : R2 → R2 are linear transformations,
(a) T (x1 , x2 ) = (1 + x1 , x2 )
(b) T (x1 , x2 ) = (x2 , x1 )
(c) T (x1 , x2 ) = (x21 , x2 )
(d) T (x1 , x2 ) = (sin(x1 ), x2 )
2. Find the range, rank, null space and nullity for the zero transformation and the identity
transformation on a f.d.v.s. V.
3. Is there a linear transformation T : R3 → R2 such that T (1, −1, 1) = (1, 0) and
T (1, 1, 1) = (0, 1)?
4. If α1 = (1, −1), α2 = (2, −1), α3 = (−3, 2), β1 = (1, 0), β2 = (0, 1) and β3 = (1, 1). Is
there a linear transformation T : R2 → R2 such that T αi = βi for i = 1, 2, 3?
5. Let F be a subfield of the complex numbers and let T be the function from F3 → F3
defined by T (x1 , x2 , x3 ) = (x1 − x2 + 2x3 , 2x1 + x2 , −x1 − 2x2 + 2x3 ).
(a) Verify that T is a linear transformation.
(b) If (a, b, c) is a vector in F3 , what are the conditions on a, b and c that the vector be
in the range of T ? What is the rank of T ?
(c) What are the conditions on a, b and c that (a, b, c) be in the null space of T ? What
is the nullity of T ?
6. Let V be an n-dimensional vector space over F and let T be a linear transformation
from V into V such that the range and null space of T are identical. Prove that n is even.
(Can you give an example of such linear transformation T ?)
7. Let T and U be the linear operators on R2 defined by T (x1 , x2 ) = (x1 , x2 ) and
U (x1 , x2 ) = (x1 , 0)
(a) How would you describe T and U geometrically?
(b) Give rules like the ones defining T and U for each of the transformations (U + T ),
U T , T U , T 2 and U 2 .
8. Let T be the linear operator on R3 defined by T (x1 , x2 , x3 ) = (3x1 , x1 −x2 , 2x1 +x2 +x3 ).
Is T invertible? If so, find a rule for T −1 like the one which defines T . Prove that
(T 2 − I)(T − 3I) = 0.
9. Find two linear operators T and U on R2 such that T U = 0 but U T ̸= 0.
10. Let V be a vector space over the field F and T a linear operator on V. If T 2 = 0,
what can you say about the relation of the range of T to the null space of T ? Give an
example of a linear operator T on R2 such that T 2 = 0 but T ̸= 0.
11. Let A be an m × n matrix with entries in F and let T be the linear transforma-
tion from Fn×1 into Fm×1 defined by T (X) = AX. Show that if m < n it may happen
that T is onto without being non-singular. Similarly, show that if m > n we may have T
is non-singular but not onto.
12. Let V be a finite dimensional vector space and let T be a linear operator on V.
Suppose that rank(T 2 )=rank(T ). Prove that the range and null space of T are disjoint,
i.e., have only the zero vector in common.
[ C be the complex vector space of 2 × 2 matrices with complex entries. Let
2×2
13. Let ]
1 −1
B= and let T be the linear operator on C2×2 defined by T (A) = BA. What
−4 4
is the rank of T ? Can you describe T 2
14. Let T be a linear operator on Fn , let A be the matrix of T in the standard or-
dered basis for Fn , and let W be the subspace of Fn spanned by the column vectors of A.
What does W have to do with T ?
15. Let T be the linear operator on C2 defined by T (x1 , x2 ) = (x1 , 0). Let B be the
′
standard ordered basis for C2 and let B = {α1 , α2 } be the ordered basis defined by
α1 = (1, i) and α2 = (−i, 2).
′
(a) What is the matrix of T relative to the pair B and B ?
′
(b) What is the matrix of T relative to the pair B and B?
′
(c) What is the matrix of T in the ordered basis B ?
(d) What is the matrix of T in the ordered basis {α2 , α1 }?
16. Let T be the linear transformation from R3 into R2 defined by T (x1 , x2 , x3 ) = (x1 +
x2 , 2x3 − x1 ).
′
(a) If B is the standard ordered basis for R3 and B is the standard ordered basis for R2 ,
′
what is the matrix of T relative to the pair B and B ?
′
(b) If B = {α1 , α2 , α3 } and B = {β1 , β2 }, where α1 = (1, 0, −1), α2 = (1, 1, 1), α3 =
(1, 0, 0), β1 = (0, 1) and β2 = (1, 0), what is the matrix of T relative to the pair B
′
and B ?
