dnullpce N(T)-
T(a (ueU:T(a)
transformation T
the
linear then from Uinto
the null space of T
V,
ifweregardof Uinto
Put
hoomorphism
sace
kernelofT LinearTransformation -
aof a
from
Rank and
Nullitytransformation
dimensional vector
space U(F)
linear
LetTbea U(F)isafinite |vectorspace.
Then
of T and is denoted by the:
where rank
sceV(F), Tis calledthe ank of T = dim R(T)
of
by o)ie
therange P(T)=
thedimension of the null space of 1, dexshp
of themulliyof Tis definedassthe
and
as wT) i.e.,
v(T) = nullity of T =dim N(T).
NUMERICAL PROBLEMS
mapping
Prob.30. Show that the defined as below
f: V{R) ’V(R),
linear.
fa, b, c) = (c, a + b) is
[R.GPV., June 2003 (I-Sen
Sol. Let a =(a,, bË, C1)) and ß=(a, bz, C) be any two elements of VR
Also, letaeF
Now f(aa. + bB) = f(a(a,, bj, C) + b(az, b, c,))
=f{(aa, ab,, ac;) + (baz, bb,, be)
= f(aa, + ba, ab + bb,, ac +bo)
= (ac, + bc,, aa, + ba, + ab, +bb)
= (ac, aa, t ab) + (bez, ba, +bb,)
a(c, a, + b) + b (cz, a, t b)
= af(a,, b, c;) + bf(a, b, c)
= af(a) +bf(p). bef
f(ac +bß) = af(a) +bf(P), va, BeV,(R); a,
Hence fis linear.
Otherwise - f(a +B) =f{(a,, b,, Ci) +(az, bz,c)}
(a, t a,, b, + b,,Ctog) t b))
- {(c t c), (aj + a,) +(b,
t by)
(C, a, t t b) +(cz, ag
= fla, b, ci) + fla bz C)
f(a) + f(B)
i.e., f(a + B)= f(a)+ f(B).
hian2024)
Vector Spaces 239
Rac)ffa (a;, bË C)) f(an,, ab, ac,)
Also, (ac), aa ab)a (Cj, a, +b)
af(a,, b,, c) af(a)
f(aa) af().
Hencefislinear.
V,(R) - Proved
Showthat the mapping f: VR) ’ V(R) defined as below
Prob31. fla, b) = (a, b, 0)
ranformationform V;(R) onto V;(R).
&e
linear (R.GPV, Dec. 2001(III-Sem)l
(a, b,)and[B=(a, b,) be any two elements of V,(R) and
Sol.Let a =
alsoleta,
beE
fa + B) = f[(a,, b) +t (az,b,,b,)] = fa, t az, b, +b)
Now = (a, + a,, b, + 0)
bT)
- (a, bj, 0) + (a,, b, 0) = f(a,, b,) + f(ah,
= f(a) + f(B).
f(a (a, b,)) = flaa,, ab) = (aa,, ab,, 0)
flaa)= af(a).
Also
=a(aj, bj, 0) = af(a,, b) = Proved
Hencefis linear.
mapping f: V;(R) ’ V,(R) defined by
Prob.32. Show that the GPV, Dec. 2006 (II-Sem)/
[R.
t c) is linear.
fla, b, c) = (a- b, a
defined as above is said to be linear, if
V,(R)
SoL.Here f:V;(R) ’
f(a+B) = f(a) +f(B) e V;(R)
f(aca) = af(a), va eF, a,ß
(a,, ay, a,), B = (b, b,, b,) and a e F.
a=
b,)]
Now f(a+p) = f[(a, a,, a,) + (b,azb,+ b,)
= f(a, + b,, a, + b, + b,) + (a, + b,)]
= [la, + b) - (a, + b,), (a,
= [(a, - a), (a, + a,)]+
[(b,- b,), (b, + b))
b) = f(a) + f(B)
= fla,, a,, a,) + f(b,, b,, ..)
f(a +B) = f(a) + f(B)
Also f(aa) = f[a (a,, a,, a)] = f(aa,, aaz, aã,)
a,, a,ta)
-(aa,- aa,, aa, + aa,) a (a, -
= af(a,, a,, a,) = af(a) ..)
f(ac) = af(a)
From relations (i) andI(ii), it is clear that the given mappingis linear.
Proved
Prob.33. Show that the
f: V(R)’ V(R)mapping
defined as
a) is linear transformation.
Sol
r V(R)-> V,(R)
defined as (R.GPV., BTech.
above is said t
tac) af(o), vaeE, a, BeV,(R)
and a-(a, ay, ay),
Let
B= (b. b, by) and aeE.
fa +B) f((a, az, ay)+ (by, bz,
Now
-fa, + b, a, + b, a, + b.)b,))
-{(a+ b)- (a, +b,),
+
(b- (art
-f(aj, az, ay) +f(b, bz,b,, b)b,=bi-
- (aj- az, aj - ay) +
b
Also
fla + B) = f(a) + f(B).
faa) fla (a, az, ay)])= (aa, aa,, aa,)
-by)
=(aa, - aaz, 2a,-
aa,) =
=af(a,, a, ay) = af(a) a
f(ac) = af(a).
Hence, f is linear.
(aj-a a1-a)
Prob.34. Show thatthe transformation
by fs, y) =(x+ 2, y +3) is not linear. f:V{R) ’V{R)Prdefoveadet
Sol. Let a = (X1, y) and ß
of V,(R) and aeF.
IR.GP.V., Dec. 2002 (II Sem
=(X, y) be any two
Now, f(a + B) = f[(X1, y) t
= (x t X, t 2,
ele rments
+ (x1 +2, yË t y1 t y2 + 3)
+ f(xz, y2) 3) + (Xt 2, y2 +3) +i,)
+ f(a) +
f(a + B) # f(a) + f(B)
f(B)
Hence f is not linear.
Prob.35. Show that the function T: R' Provel
’ R by defined -
is a linear
transformation.
Sol. Let a =(x, Xz, Xg), ß
=
[R.GPV, May/June 2006 (1I-Sen
If a, b e R,
then (y1 y2» y3) e V, (R)
T(aa +bß) =T[a (X1, Xy,
=T(ax +byj, Xi) +b (y, y2> Y))
ax, +by, ax, +by)
-[(ax,
=(a
+ by) --(ax,
(ax, + by), (ax, t by) +(ax, +byl
-[a (x1-x)
+ b(y1-
y. a (x t+t xy) +b(y,t l
(x1-X),
a(x1 t x)] +[b (y- Y
=a ((x- x), (X1 t
:. Tis a =aT(x1, Xz, X¡) + t x)] +b(y-aT(a) ), )8+bTB)
linear bT(y1: y2, y)
transformation
from R3 into R2.
Vector Spaces 241
map T: R- R' given by
Showthar the linear.
Prah36
(R.GPV, May 2019)
(K x).B: ) e
V(R)
a -
Sol LetR.then
Ifabe
y)
x) +b(y by,)
T(aa bB) -T[a(x,,
-T(ax, 4+ by,, ax,+
-((ax, +by,) +(ax, + by). (ax, +by,)
- (ax, + by,). (ax, + by)
- [a(x, + x) + b(y,t Y), a(x, - x) + b(y,- Y). (ax, + by,))
= aT(x,, x) + bT(Y Y)
=aT(a) + bT(B) Proved
from R, into Rq.
:Tisa inear transformation over the same field F.
and Vhe two vector subspaces
Prob.37. Let U TT:U+ Vis a linear transformation iff-
that
afunction
1(),for all u, veUand a eF
Show v) =a T(u) + June 2007 (LII-Sem)]
Tlau + (R.GP.,
any u, ve Uand a e E.
SoL Consider transformation.
Let T be a linearT(au) = aT(u)
So
Consequently
+ v) = T(au) + T(v) = aT(u) + T(v)
T(au
given condition
Conversely, let T satisfy the
Then T(u+ v)= T(1.u + v) T(u) + T(v)
=1.T(u) + T(v) =
vector
We know that
transformation from a vector space U(F) into a
IfTis alinear
pace V(F).
Then
T(0) = 0
So that T(au) = T(au + 0) aT(u)
+0=
= T(au) + T(0) = aT(u) Proved
Hence T is a linear transformation.
defined by
Prob,38. Show that the mapping f: ,(R) ’ V;(R) linear mapping
a
a) =(3a,- 2a, +Fag a, - 3a, - 2a) is
V(R) into V(R).
Sol. f: V,(R) ’\ V,(R) defiend as above is said to be linear, if
f(a + B)= f(a) + f(B)
f(aa) = af(a), vaeF; a, B eV;(R)
Purting K112 y
11
From equations(i). (i)and (iv), we obtain the coefficient
A -34 -V2] 11/4 35/4 11/2
Bs/4 15/4 -7/2and its transpose is-3/4 15/4
W2 -3/2 0 -1/2 -7|2 A
Prob.42,/Showthat T: Vz (R) ’ V3(R) is defined as
TYa, b) =(a-6, b- 4, - a) is linear
transformation.
Sol We are given that a linear
transformation (R.GPV.
T:VaR) , Nov. 10
by
T(a, b) = (a - b, b -- a, - a)
-’V,Ridefp
We need to find the nullity of linear transformation T.
Let (a, b) e ker T
Then, T(a, b) = (0, 0, 0)
Now using the definition of linear
(a- b, b - a, - a) = (0, 0, 0) transformation,
we get
Comparing the components of the coordinates, we get
a-b= 0
b-a= 0
-a= 0
Solving for a and b, we get
a= 0, b=0
Therefore, ker T = {(0, 0)}
Thus, dim(ker T) =0, i.e., Tisx
nullity of linear transformation
