Text:
Linear Algebra Done Right
by Sheldon Axler, Springer 2nd ed..
Exam
Schedule:
Midterm 1: Monday, Oct. 22, in class
Midterm 2: Monday, Nov. 19, in class
Final: Wednesday, December 12, 811am, in class
Grading:
Homework 10%; Midterms 20% each; Final 50%.
HW #1 (due 10/5)
Batch 1:
1) Verify that <x, y>_k (you decide what `k'
stands for :))=nx_1y_1+(n1)x_2y_2 + ... + x_ny_n
defines an inner product on R^n. Then compute the norm
of
u=(a_1, a_2, ..., a_n).
2) Given real numbers a_ij, i, j=1, 2, define <x,
y>=a_11 x_1 y_1 + a_12 x_1 y_2 + a_21 x_2 y_1 + a_22
x_2 y_2, determine the conditions that make <x, y>
an inner product on R^2.
3) Given an inner product on R^2, compute xy. Hence
use the cosine law to deduce that <x, y>=x
y cos \theta, with \theta =the angle between x and
y, when <x, y> is the dot product.
Batch 2:
p122: 2, 6.
Batch 3:
p122: 3, 4, 5.
Solutions
HW #2 (due 10/12)
Batch 1:
p122: 7, 8(hint: given v, how would you define
<u, v>? you need to check 1). positivity; 2)
linear in first slot; 3) symmetry/conjugate symmetry.
You can skip the linearity), 9, 13.
Batch 2:
1) For V=R^3, <x, y>=dot product, v_1=(1, 0, 1),
v_2=(0, 1, 1), v_3=(1, 3, 3), apply the GramSchmidt to
obtain an orthonormal basis e_1, e_2, e_3. Then write
v=(1, 1, 2) in terms of e_1, e_2, e_3.
2) For V=R^2, <x, y>=2 x_1 y_1  x_1 y_2  x_2 y_1
+ 2 x_2 y_2, apply GS to v_1=(1, 0), v_2=(0, 1) to
obtain an orthonormal basis e_1, e_2.
p123: 10, 11.
Batch 3:
1). Prove that U^\perp is always a subspace of V.
2). For V=R^3, <x,
y>=x_1y_1+x_2y_2x_1y_3x_3y_1+4x_3y_3, extend the
orthonormal list e_1=(1,0, 0), e_2=(0, 1, 0) to an
othonormal basis.
p124: 16.
Solutions
HW #3 (due 10/19)
Batch 1:
p124: 21, 22, 23.
Batch 2:
1) Let V be an inner product space (real or complex) and
e_1, e_2, ..., e_n an orthonormal basis. Show that for
any u, v in V
<u, v>=<u, e_1>\overline{<v, e_1>} +
... + <u, e_n>\overline{<v, e_n>} (Here
\overline{ } means the complex conjugate; for real
vector space, it does nothing)
What happens when u=v?
p125: 27, 28.
Solutions
Solutions
to Midterm I
HW #4 (due 10/26)
Batch 1:
p124: 19, 29, 30.
Batch 2:
1). Verify a) (T_1+T_2)^*=T_1^* + T_2^*; b)
(aT)^*=\bar{a} T^*; (ST)^*=T^*S^*.
p125: 30, 31.
Batch 3:
p125: 24.
Solutions
HW #5 (due 11/2)
Batch 1:
p158: 1, 2, 3(a).
Batch 2:
p159: 8, 15. Plus
1). Consider T: R^2 > R^2 given by T(x)=Ax where
A = matrix( 0, 1 \\ 1, 0). Show that T^2+I=0. Compare
this with Lemma 7.11.
2). Consider T: R^2 > R^2 given by T(x)=Ax where
A = matrix( 1, 1 \\ 0, 1). Show that U={x_2=0} is
invariant under T but U^{\perp} is NOT.
Batch 3:
1). Determine if the following matrices can be
diagonalized. A_1=matrix( 1, 2, 3, 4, 5\\ 2, 2, 3, 4,
5\\ 3, 3, 3, 4, 5\\4, 4, 4, 4, 5\\ 5, 5, 5, 5, 5)
A_2= matrix(2, 0, 0, 0, 0\\1, 2, 0, 0, 0\\2, 1, 2, 0,
0\\1, 2, 1, 2, 0\\2, 1, 2, 1, 2).
2). Let A be a symmetric real matrix and \alpha, \beta
real numbers such that A^2+ \alpha A + \beta I=0.
(a). Show that \alpha^2 >= 4\beta.
(b). Show that there are real numbers \lambda_1,
\lambda_2 such that (A \lambda_1 I)(A \lambda_2 I)=0.
(c). Deduce that one of the \lambda_1, \lambda_2 is an
eigenvalue of A.
Solutions
HW #6 (due 11/9)
Batch 1:
1). Let C^2 be given the inner product <z, w>=4z_1
\bar{w_1} + 2z_1 \bar{w_2}+ 2z_2 \bar{w_1}+ 2 z_2
\bar{w_2} and T: C^2 > C^2 be given by T(z)=Az
where A = matrix( 2, 3/2 \\ 0, 1). Determine if T is
self adjoint (with respect to this given inner product).
2). Same for A = matrix( 1, 0 \\ 0, 2).
3). Given a complex vector space V and a linear map T: V
> V, define T_1=1/2(T+T*), T_2=1/(2i)(TT*).
(a). Show that both T_1, T_2 are self adjoint.
(b). Show that T is normal if and only if T_1T_2=T_2T_1.
Batch 2:
1). Verify that for any linear map T on a complex inner
product space V, <Tu, v>=1/4[ <T(u+v), u+v>
 <T(uv), uv> + i<T(u+iv), u+iv> 
i<T(uiv), uiv>].
2). Let V be an inner product space (real or complex),
T_1, T_2 are linear maps on V such that T_1T_2=T_2T_1.
Show that null(T_1\lambda I) is invariant under T_2.
3). Same as in 2). In addition, T_1, T_2 are self
adjoint. For simplicity, assume that T_1 has only two
distinct eigenvalues \lambda_1, \lambda_2.
(a) Use Corollary 7.14 to deduce that
V=null(T_1\lambda_1 I) \oplus null(T_1\lambda_2 I).
(b) Use 2) to get T_2:
null(T_1\lambda_1 I) > null(T_1\lambda_1 I) and
T_2: null(T_1\lambda_2 I) > null(T_1\lambda_2
I). Hence obtain an o.n. basis
e_1, ... , e_k of null(T_1\lambda_1 I), and o.n. basis
\tilde{e}_1, ..., \tilde{e}_l of null(T_1\lambda_2 I).
(c) Conclude that there is an o.n.
basis of V which are eigenvectors of both T_1 and T_2.
Batch 3:
p158: 6; plus
1). Let V be an real inner product space and T: V
> V self adjoint. Derive a formula for <Tu,
v> in terms of <T(u+v), u+v> and <T(uv),
uv>. Hence deduce that
<Tv, v>=0 for all v in V implies that T=0. Give an
example when this fails if T is not self adjoint.
Solutions
HW #7 (due 11/16)
Batch 1:
1). Let T be a linear map on V and U a subspace of V.
(a) Show that U is invariant under T if and only
if there is a basis v_1, ... , v_m of U such that Tv_1,
... , Tv_m are all in U;
(b) Conclude that U=span(u_1, ..., u_k)
where each u_1, ..., u_k is an eigenvector of T, is
invariant under T;
(c) What is a one dimensional invariant subspace?
2). Let V be a complex inner product space of dim 3, T
is a normal map on V and U an invariant subspace of T of
dim 2. Show that U^{\perp} is also invariant.
3). Same as above but dim V=4.
Batch 2:
p159: 17, 18; Plus
1). Recall that for a linear map T on a complex inner
product space, T=T_1+iT_2, with T_1=1/2(T+T*) and
T_2=1/(2i)(TT*) both self adjoint. Moreover, T is
normal if and only if T_1T_2=T_2T_1.
(a) If u is an eigenvector of both
T_1, T_2, show that u is also an eigenvector of T.
(b) Give another proof of the Complex
Spectral Theorem II by using the Complex Spectral
Theorem I and HW#6.
Solutions
Solutions
to Midterm II
HW #8 (due 12/3, postponed to
12/5)
Batch 1:
p188: 3, plus
1). Find all generalized eigenvectors of T: C^4
> C^4 given by T(x)=Ax where A = matrix( 2, 2,
2, 2 \\ 0, 2, 2, 2 \\ 0, 0, 2, 2 \\ 0, 0, 0, 2).
2) Use the definition of generalized eigenvectors to
show that the set of all generalized eigenvectors with
eigenvalue \lambda is a subspace.
Batch 2:
p188: 5, 6, 7.
Batch 3:
1). Let U be a complex vector space and N: U > U a
nilpotent operator. Put V=range N. Assume that there are
vectors v_1, v_2, v_3 in V such that
v_1, Nv_1, v_2, Nv_2, v_3, Nv_3, N^2v_3 gives a basis of
V and Nv_1, Nv_2, N^2v_3 form a basis of null N \cap V.
(a) Show that there are vectors u_1, u_2, u_3, ... , u_k
(k>=3) such that u_1, Nu_1, N^2u_1, u_2, Nu_2,
N^2u_2, u_3, ..., N^3u_3, u_4, ... , u_k form a
basis of U
and N^2u_1, N^2u_2, N^3u_3, u_4, ... , u_k gives a basis
of null N.
(b) Find the Jordan form of N in this basis.
Solutions
