MIDTERM 1 FOR ADVANCED LINEAR ALGEBRA
Date: Wednesday, Nov 8, 2000 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1.
(i) [5%] Give an example of a nondegenerate symmetric bilinear form of Witt index 1 on a two- dimensional real vector space.
(ii) [5%] Give an example of nonzero quadratic form on a two-dimensional real vector space.
2. Let H be the quaternion algebra over R and let M2(R) be the matrix algebra of two by two real matrices.
(i) [5%] Is H isomorphic to M2(R) as vector spaces over R? Why or why not?
(ii) [5%] Is H isomorphic to M2(R) as R-algebras? Why or why not?
3. Let V = F2 be a two-dimensional vector space over a field F . Define two unit vectors i = (1, 0) and j = (0, 1). It is obvious that {i, j} is a basis for V . Define f : V × V → F by
f ((a1, b1), (a2, b2)) = 2b1b2
for a1, a2, b1, b2∈ F .
(i) [5%] What is the radical of V ?
(ii) [5%] Suppose that the characteristic of F is not 3. Find the matrix presenting the form f with respect to the basis {9i, 3j}.
4. [10%] Find the Jordan canonical form of the matrix
0 1 1 0 0 1 0 0 0
.
5. Let V = R2 be a two-dimensional real vector space. Fix a basis {i, j} for V where i = (1, 0) and j = (0, 1). Define
h(x1, y1), (x2, y2)i = x1x2+ y1y2
for x1, x2, y1, y2∈ R.
(i) [5%] Show that the linear transformation
"
cos θ sin θ
− sin θ cos θ
#
where θ is a real number is an isometry of V .
(ii) [5%] Suppose that τ : V → V is an isometry. Show that aτ for a ∈ R is an isometry if and only if a = ±1.
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2 MIDTERM 1 FOR ADVANCED LINEAR ALGEBRA
6. Let F be a field and V be a vector space over F . Suppose that f is a bilinear form on V . (i) [5%] Show that if f is alternating, then f is skew-symmetric.
(ii) [5%] Show that if f is skew-symmetric and the characteristic of F is not 2, then f is alternating.
7. Let V = C2 be a two-dimensional vector space over C. Define a symmetric bilinear form f on V by f ((x1, x2), (y1, y2)) = x1y1+ x2y2
for x1, x2, y1, y2∈ C.
(i) [5%] Show that f is isotropic.
(ii) [5%] Show that V is a hyperbolic plane.
8. Let V = R2be a two-dimensional vector space over a field R with a nondegenerate skew-symmetric bilinear form h, i defined by
h(x1, y1), (x2, y2)i = x1y2− y1x2
for x1, x2, y1, y2∈ R.
(i) [5%] Suppose that f is a linear functional on V . The Riesz representation theorem tells us that there is a unique element x ∈ V such that f = φx where φx∈ V∗ is defined by φx(v) = hv, xi.
Find x for the linear functional f defined by f ((x, y)) = 2x + 3y for x, y ∈ R.
(ii) [5%] Let S be the subspace spanned by the vector (1, 0). Is V = S ⊥ S⊥? Why or why not?
9. [10%] Let P2⊂ R[x] be the space of polynomials of degree less than or equal to 2. Define hf, gi =
Z 1
−1
f (x)g(x) dx
for f, g ∈ P2. We know that {1, x, x2} is a basis for P2. Apply Gram-Schmidt orthogonalization process to {1, x, x2} to find an orthogonal basis for P2.
10. Let `2be the set of all real infinite sequences (an) such thatP∞
n=1|an| is finite. Define (an) + (bn) = (an+ bn) and r(an) = (ran) for (an), (bn) ∈ `2 and r ∈ R.
(i) [5%] Show that `2 is a vector space over R.
(ii) [5%] Show that `2 is an inner product space under the inner product h, i defined by h(an), (bn)i =
X∞ n=1
anbn.
