Download Algebra I

Algebra I
Midsemester Examination
(25/9/2006)
1. (a) Show that if B1 =< x, y, z > is a basis for V ,then B2 =< x +
y, y + z, z + x > is also a basis for V What are the coordinates of
x + y + z with respect to B1 ?with respect to B2 ?
(b) Let L : V → V map B1 to B2 .What is the matrix of L with
respect to B1 ?with respect to B2 ?Suppose you choose basis B1
for the domain and B2 for the range.Then what is the matrix of
L?Suppose instead you choose basis B2 for the domain and B1 for
the range.Then what is the matrix of L?
0
0
(c) Suppose B2 =< f, g, h > is the basis for V dual to B2 .What is
f (x)?g(y)?
(d) Let U be the subspace of V spanned by x and y.What is the
dimension of U ◦ ?Describe the coordinates of a basis of U ◦ in terms
0
of B2
(e) Let L be as in (ii) and A : V → V be any linear map.Show that
LA is similar to AL.Is it true that for any A, B : V → V linear
maps, AB is always similar to BA
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2. (a) Let F be any field.Let m, n be positive integers.Let f1 , f2m be
0
elements of (F n ) .Consider the map
L : Fn → Fm
x → Lx = (f1 (x), . . . , fm (x))
Show that L is linear.Conversely show that every linear map from
F n to F m arises this way
(b) Let V be a finite dimensional vector space over F .Let f, g be in
0
V .Suppose kernel f ⊂ kernel g.Show that g = af for some a in F
(c) Let V be n dimensional.Show that there is a one to one correspondence
between m dimensional subspaces and (n−m) dimensional subspaces
of V
(d) Let F be an infinite field and V as in (ii).Let x1 , . . . , xT be any
0
finite collection of nonzero vectors.Show that there is an f in V
such that each f (xi ) 6= 0.
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3. (a) Denote by RC m the ’decomplexification’ of C m ,ie,C m considered
as a real vector space.Show that if B =< u1 , . . . , um > is a
basis for C m ,then RB =< u1 , . . . , um , iu1 , . . . , ium > is a basis
for RC m
Let A : C m → C n be a C linear map.Then the ’decomplexification’
of A is the R linear map RA : RC m C n which coincides with
A pointwise.Let B1 and B2 be bases for C m , C n respectively
, and let (A) = ((α) + i(β)) be the matrix of A with respect
to them.What is the matrix of RA with respect to RB1 , RB2 ?
(b) Denote by CRm the ’complexification’ of Rm constructed as follows:the
elements of CRm are (u + iv),u, v in Rm . The complex vector
space structure is
(u1 + iv1 ) + (u2 + iv2 ) = (u1 + u2 ) + i(v1 + v2 )
a, b ∈ R, (a + ib)(u + iv) = (au − bv) + i(bu + av)
Show that if B =< u1 , . . . , um > is a basis for Rm ,then
CB =< u1 + i0, . . . , um + i0 > is a basis for CRm
Let A : Rm → Rn be an R linear map.Then the ’complexification’
of A is the C linear map CA : CRm Rn defined by CA (u + iv) =
Au+iAv.LetB1 and B2 be bases for Rm , Rn respectively , and
let (A) be the matrix of A with respect to them.What is the
matrix of CA with respect to CB1 , CB2 ?
(c) Consider the 2n dimensional real vector space RCRn obtained
from Rn by complexification followed by decomplexification
This space contains an n dimensional subspace containing
vectors of the form u + i0, u ∈ Rn ,called the ’real plane’.The
’imaginary plane’ is the subspace of vectors of the form 0 +
iv, v ∈ Rn .RCRn is the direct sum of these two subspaces
The map iI of multiplication by i in CRn is transformed after
decomplexification into an R linear map R(iI) .What is the
image of the real plane under this map?The image of the
imaginary plane?What isR(iI) .R(iI) ?
Let < u1 , . . . , un > be a basis for Rn and < u1 , . . . , un , iu1 , . . . , iun >
be a basis for RCRn .What is the matrix of R(iI) in this basis?
Let σ : RCRn → RCRn be the map σ(u + iv) = u − iv denoted
by (u + iv).What is σ.σ?What is the matrix of σ in the above
basis?
Let A : CRm Rn be a C linear map.By the complex conjugate A
of A is meant the map A : CRm Rn , Ax = Ax.Show that A is
C linear.
2
Show that a C linear map A : CRm Rn is the complexification
of a real map if and only if A = A.Is the map iI above the
complexification of a real map.?
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