Download MTL101:: Tutorial 3 :: Linear Algebra

MTL101:: Tutorial 3 :: Linear Algebra
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Notation: F = R or C, Pn := {f ∈ F[x] : deg f < n}

 
t
1 2 3
1 0 2 5
Find a basis of the row space of the following matrices: 2 3 4, 0 3 1 1 .
3 4 5
3 1 0 1
For i ∈ {1, 2, . . . , m}, define pi : Fm → F by pi (x1 , x2 , . . . , xm ) = xi (the i-th projection).
(a) Show that it is a linear transformation.
(b) If T : Fm → F is a linear transformation then it is an F-linear combination of the projections, that is, T = a1 p1 + a2 p2 · · · + am pm for a1 , . . . , am ∈ F.
(c) Further, show that S : Fm → Fn is a linear transformation if and only if for each i ∈
{1, 2, . . . , n}, the composition S ◦ pi : Fm → F is a linear transformation.
(d) If S : Fm → Fn is a linear transformation then S(x1 , x2 , . . . , xm ) = (y1 , y2 , . . . , yn ) where
yi = ai1 x1 + ai2 x2 + · · · + aim xm for aij ∈ F with (1 ≤ i, j ≤ m).
Find the rank and nullity of the following linear transformations. Also write a basis of the
range space in each case.
(a) T : F3 → F3 defined by T (x, y, z) = (x + y + z, x − y + z, x + z).
(b) Assume that 0 ≤ m ≤ n. T : Fn → Fm defined by T (x1 , x2 , . . . , xn ) = (x1 , x2 , . . . , xm ).
Write the matrix representations of the linear operators with respect to the ordered basis B.
(a) T : R2 → R2 where T (x, y) = (x, y), B = {(1, 1), (1, −1)}.
(b) D : Pn+1 → Pn+1 such that D(a0 + a1 x + · · · + an xn ) = a1 + 2a2 x + · · · + nan xn−1 ,
B = {1, x, . . . , xn }. x y
x+w z
1 0
0 1
0 0
0 0
(c) T : M2 (F) → M2 (F),
7−→
,B ={
,
,
,
}.
z w
z+w x
0 0
0 0
1 0
0 1
Suppose dimV = dimW < ∞ and T : V → W is a linear transformation. Show that the
following statements are equivalent
(a) T is an isomorphism.
(b) T is injective (i.e., one to one).
(c) kerT = 0.
(d) T is surjective (i.e., onto).
Suppose m > n. Justify the following statements:
(a) There is no one to one (injective) R-linear transformation from Rm to Rn .
(b) There is no onto (surjective) R-linear tranformation from Rn to Rm .
Find the eigenvalues, eigenvectors and dimension of eigen-spaces of the following operators.
(a) T : R2 → R2 with T (x, y) = (x + y, x),
(b) T : R2 → R2 with T (x, y) = (y, x),
(c) T : R2 → R2 with T (x, y) = (y, −x)
(d) T : C2 (C) → C2 (C) with T (x, y) = (y, −x).
(e) Cn → Cn defined by (x1 , x2 , . . . , xn ) 7→ (xn , x1 , . . . , xn−1 ).
(f) C2 → C2 defined by (z1 , z2 ) 7→ (z1 − 2z2 , z1 + 2z2 ).
(To be discussed in the lecture class) Suppose {λ1 , . . . , λt } is the complete set of distinct
eigenvalues of a linear operator T : V → V . Denote by Vi the eigenspace of λi . If Bi is a
t
t
P
basis of Vi then show that ∪ Bi is a basis of the sum Vi . Conclude that it is a direct sum.
i=1
Further, show that T is diagonalizable if and only if
i=1
t
P
Vi = V . Equivalently, show that T is
i=1
diagonalizable if and only if dimV = dimV1 + dimV2 + · · · + dimVt .
(9) Find a basis B such that [T ]B is a diagonal matrix in case T is diagonalizable. Find P such
that [T ]B = P [T ]S P −1 where S is the standard basis in each case.
(a) T : C2 → C2 defined by T (x, y) = (y, −x).
(b) T : R3 → R3 defined by T (x, y, z) = (5x − 6y − 6z, −x + 4y + 2z, 3x − 6y − 4z).
(c) T : C2 → C2 defined by T (x, y) = (x cos θ + y sin θ, −x sin θ + y cos θ).
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(10) Characteristic polynomial of a matrix is satisfied by the matrix (Cayley-Hamilton). Use it to
find (invertibility and) the inverse of the following linear operators.
i) (x, y, z) 7→ (x + y + z, x + z, −x + y).
ii) (x, y, z) 7→ (x, x + 2y, x + 2y + 3z).
(11) Which of the following is an inner product.
(a) h(x1 , y1 ), (x2 , y2 )i = x1 x2 + y1 y2 + 3 on R2 over R.
(b) h(x1 , y1 ), (x2 , y2 )i = x1 x2 − y1 y2 on R2 over R.
(c) h(x1 , y1 ), (x2 , y2 )i = y1 (x1 + 2x2 ) + y2 (2x1 + 5x2 ) on R2 over R.
(d) h(x1 , y1 ), (x2 , y2 )i = x1 x2 + y1 y2 on C2 over C.
(e) h(x1 , y1 ), (x2 , y2 )i = x1 x̄2 − y1 ȳ2 on C2 over C.
(f) If A, B ∈ Mn (C) define hA, Bi = Trace(AB̄).
(g) Suppose C[0, 1] is the space of Rcontinuous complex valued functions on the interval [0, 1]
1
and for f, g ∈ C[0, 1], hf, gi := 0 f (t)ḡ(t) dt.
a b
(12) Suppose A =
∈ M2 (R) is such that a > 0 and det(A) = ad − b2 > 0. Show that
b d
hX, Y i = X t AY is an inner product on R2 .
p
(13) Suppose V is an inner product space. Define ||v|| := hv, vi. Show the following statements.
(a)
(b)
(c)
(d)
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||v|| = 0 if and only if v = 0
For a ∈ F , ||av|| = |a|||v||.
||v + w|| ≤ ||v|| + ||w||.
|||v|| − ||w||| ≤ ||v − w||.
hv, wi = 0 ⇐⇒ ||v + w||2 = ||v||2 + ||w||2 .
(14) Use standard inner product on R2 over R to prove the following statement: “A parallelogram
is a rhombus if and only if its diagonals are perpendicular to each other.”
(15) Find with respect to the standard inner product of R3 , an orthonormal basis containing (1, 1, 1).
(16) Find an orthonormal basis of
R 1P3 := {f (x) ∈ R[x] : deg f (x) < 3} with respect to the inner
product defined by hf, gi := 0 f (t)g(t)dt.
(17) Suppose W is a subspace of the finite dimensional inner product space V . Define W ⊥ := {v ∈
V : hw, vi = 0 ∀ w ∈ W }. Show the following statements.
(a)
(b)
(c)
(d)
W ⊥ is a subspace of V .
W ∩ W⊥ = 0
V = W ⊕ W ⊥.
(W ⊥ )⊥ = W .
(18) Suppose W = {(x, y) ∈ R2 : x+y = 0}. Find the shortest distance of (a, b) ∈ R2 from W with
respect to i) the standard inner product, ii) the inner product defined by h(x1 , y1 ), (x2 , y2 )i =
2x1 x2 + y1 y2 .
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