MA 102 : Linear Algebra and Integral Transforms Tutorial Sheet - 3
Second Semester of Academic Year 2018-2019
1. Determine which of the following mappings are linear.
(a) T :R3→R2,T
x y z
= xy
yz
.
(b) T :R3→R3,T
x y z
=
x+ 1 y z
.
(c) Transpose mappingT :Rm×n→Rn×m,T(A) =At. (d) Trace mappingtr:Rn×n→R,tr(A) =trace(A).
(e) The evaluation mappingεu:R[x]→R,u∈R, defined byεu(a0+a1x+· · ·+anxn) = a0+a1u+· · ·+anun, where R[x] is the set of all polynomials over R.
(f) T :R[x] → R∞, defined by T(a0+a1x+· · ·+anxn) = (a0, a1,· · ·, an,0,0,0,· · ·), whereR∞ is the set of all real sequences.
2. Find out whether the following statements are true or false.
(a) The differential mapping D:C1(I)→ C0(I), defined byD(f(x)) =f0(x) is injective, whereI is an open interval inR.
(b) The mapping defined in Q.1(f) is surjective but not injective.
(c) The evaluation mapping is surjective.
(d) The trace mapping is injective but not surjective.
3. Find Null space and Range space of the following mappings.
(a) S:Rn→Rn such thatS
x1
·
·
· xn−1
xn
=
0 x1 x2
·
· xn−1
(b) T :Rn→Rnsuch that T
x1
·
·
· xn−1
xn
=
x1 x2−x1
·
·
·
xn−xn−1
4. Let T be a linear operator on a vector space V, let v ∈ V and let m be a positive integer such thatTmv= 0 and Tm−1v6= 0. Then show that v, T v, ..., Tm−1v are linearly independent.
5. Consider the vector spaceP2(x) of polynomials with real coefficients and of order at most 2. Find the rank and nullity of the following linear transformation T :P2(x)→M2×2(R) defined byT(p(x)) =
p(1)−p(2) 0
0 p(0)
.
6. LetT :Rn→ Rn first reflects points throughx-axis and then reflects points through the liney =x. Find the mappingT.
7. Find a 2×2 singular matrix B that maps (1,1)t to (1,3)t.
8. LetU, V, W be vector spaces over K. LetT ∈L(V, W) and let S ∈L(U, V). Show that kerS ⊆ker T S and im T S ⊆im T.
9. Find a linear mapF :R3 →R4 whose image is spanned by (1,2,0,4) and (2,0,1,3).
10. (a) Give an example of linear transformations T and U such that N(T) = N(U) and R(T) =R(U).
(b) Give an example of a linear transformationT :Rn→Rn such thatN(T) =R(T).
11. LetT :P[x]→R2defined byT(p(x)) = p(1)
p(−1)
. Verify thatTis a linear transformation and also find its Kernel.
12. Let V be a finite dimensional vector space and S, T ∈L(V). Show that if ST is identity operator, then so isT S. Give an counter example showing that the given statement may not be true for infinite dimensional vector spaces.
13. Let T :R3 →R2 and let S:R2 →R3. Show that ST is not injective.
14. Consider the matrix mappingT :R4→R3such thatT(X) =AX, whereA=
1 2 3 1
1 3 5 −2
3 8 13 −3
.
Find a basis and the dimension of the image of T as well as of the kernel of T.
15. Consider the linear operatorT onR3 defined byT(x, y, z) = (2x,4x−y,2x+ 3y−z). Find formula forT2.
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