Problem Set-V
All notations are standard and are given explicitly in the last page of this sheet.
1. Suppose V is a vector space over F. Let < , >: V × V −→ F be defined as follows:
(a) h, i : Rn × Rn −→ R defined by hx̃, ỹi = x̃ỹ t = x1 y1 + x2 y2 + · · · + xn yn .
(b) h, i : Cn × Cn −→ C defined by hx̃, ỹi = x̃ỹ t .
(c) h, i : Cn × Cn −→ C defined by hx̃, ỹi = x̃ỹ ∗ .
(d) h, i : R2 × R2 −→ R defined by hx̃, ỹi = x1 y1 − 2x1 y2 − 2y1 x2 + 9x2 y2 .
(e) h, i : R3 × R3 −→ R by h(x1 , x2 , x3 ), (y1 , y2 , y3 )i = x1 y1 + x2 y1 + x1 y2 + 2x2 y2 +
3x3 y2 + 3x2 y3 + 9x3 y3 .
(f) h, i : Mn (R) × Mn (R) −→ R defined by hA, Bi = trace(AB t ).
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(g) h, i : P1 (R) × P1 (R) −→ R defined by hp(x), q(x)i = 0 p(x)q(x)dx.
Check whether the given function defines an inner product on V or not.
2. Let A be a 2 × 2 matrix with real entries. Define a map h , i from R2 × R2 to R by
y
h(x1 , x2 ), (y1 , y2 )i = x1 x2 A 1 . Show that h , i is an inner product on R2 iff
y2
T
A = A , a11 > 0, a22 > 0 and det(A) > 0.
3. Let V be a real or complex vector space with an inner product. Show that ||x − y||2 +
||x + y||2 = 2||x||2 + 2||y||2 , for every x, y ∈ V . This is called parallelogram law.
4. (a) If V is a real inner product space, then for any x, y ∈ V , we have hx, yi =
1
4
(||x + y||2 − ||x − y||2 ).
(b) If V is a complex inner product space, then for any x, y ∈ V , we have hx, yi =
1
4
(||x + y||2 − ||x − y||2 + i||x + iy||2 − i||x − iy||2 ).
5. Let V be a real inner product space.
(a) Show that x − y ⊥ x + y iff ||x|| = ||y|| (The geometric meaning of this is that a
parallelogram is a rhombus iff the diagonal are perpendicular).
(b) Let V be a real inner product. Show that x ⊥ y iff ||x − y||2 = ||x||2 + ||y||2 (This
is Pythagoras theorem and its converse).
(c) Show that if ||x + y|| = ||x|| + ||y||, one is scalar multiple of the other.
6. Apply Gram-Schmidt process to obtain an orthonormal set:
(a) {(−1, 0, 1), (1, −1, 0), (0, 0, 1)} in R3 with usual inner product
Z 1
2
(b) {1, p1 (t) = t, p2 (t) = t } of P2 (R) with inner product hp, qi = p(t)q(t)dt
0
(c) {(1, −1, 1, −1), (5, 1, 1, 1), (2, 3, 4, −1)} in R4 with usual inner product
1
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7. Let V = C([0, 1]) with inner product hf, gi = 0 f (x)g(x)dx. Find the orthogonal
complement of the subspace of polynomial functions.
8. Let V = Mn (C) with the inner product hA, Bi = tr(AB ∗ ). Find the orthogonal
complement of the subspace of diagonal matrices.
9. Let W be a subspace of a finite dimensional inner product space V and x ∈ V such
that hx, yi + hy, xi ≤ hy, yi for all y ∈ W . Show that x ∈ W ⊥ .
10. Consider the subspace W = {(x, y, z, w) | x + 2y + z + w = 0 = x + y − 2z, w = 0} of
the standard inner product space R4 . Find an orthonormal basis of W and W ⊥ .
11. Consider R4 with the usual inner product. Let W be the subspace of R4 consisting
of all vectors which are orthogonal to both (1, 0, −1, 1) and (2, 3, −1, 2). Find an
orthonormal basis of W .
12. Find the projection of v = (3 + 4i, 2 − 3i) along the vector w = (5 + i, 2i) in C2 over C.
13. 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 .
Note:
1. x̃- a vector in Fn , i.e., x̃ = (x1 , x2 , . . . , xn ).
2. AT - transpose of a matrix A.
3. A∗ - conjugate transpose of a matrix A.
4. x ⊥ y means x is orthogonal to y i.e. hx, yi = 0.
5. W ⊥ denotes orthogonal complement of W .
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6. Let f : [0, 1] −→ R be a continuous map such that 0
xn f (x)dx = 0, for every n ∈
N ∪ {0}. Then f (x) = 0 for every x ∈ [0, 1].
7. The inner product defined by h(x1 , x2 , · · · , xn ), (y1 , y2 , · · · , yn )i = x1 y1 + x2 y2 + · · · +
xn yn is called usual inner product on Rn .
8. The inner product defined by h(z1 , z2 , · · · , zn ), (w1 , w2 , · · · , wn )i = z1 w̄1 + z2 w̄2 + · · · +
zn w̄n is called usual inner product on Cn .
