Fall 2012
Math 419
Problem Set 6
Due on Thu, Oct 18
1) In Rnm , let us dene the inner product as A, B = trace(AT B ).
(a) Find a formula for this inner product in Rn1 = Rn .
(b) Find a formula for this inner product in R1m (i.e., the space of
row vectors
Theorem 1. For an n n matrix A, the following statements are equivalent. (i) A is invertible.
(ii) Ax = b has a unique solution x for b Rn . (iii) rref(A) = In . (iv) rank(A) = n. (v)
image(A) = Rn . (vi) ker(A) = cfw_0. (vii) The column vectors of A form
Fall 2012
Review Sheet for Final Exam
Read also Review Sheet for Midterm Exam
Terminology
Can you explain these words?
1. permutation
2. transposition
3. sign of a permutation
4. determinant
5. minors and cofactor
6. rotation matrices
7. classical adjoint
Fall 2012
Review Sheet for Midterm Exam
Terminology
Can you explain these words?
1. reduced row-echelon form
2. elementary row operations
3. consistent
4. rank
5. linear combination
6. linear transformation
7. orthogonal projection
8. image
9. span
10. ke
MATH 419 SECTION 001
MIDTERM
October 25, 2012,
Instructor: Manabu Machida
Name:
To receive full credit you must show all your work.
Theorems listed at the end can be used without proof.
One side of a US letter size paper (8.5" 11" ) with notes is OK.
Theorem 1. If ker(A) = cfw_0, then the linear system Ax = b has the unique least-squares solution
x = (AT A)1 AT b.
Theorem 2. The determinant of an (upper or lower) triangular matrix is the product of the
diagonal entries of the matrix.
Theorem 3. Elemen
MATH 419 SECTION 001
FINAL
December 20, 2012,
Instructor: Manabu Machida
Name:
To receive full credit you must show all your work.
Theorems listed at the end can be used without proof.
Both sides of a US letter size paper (8.5" 11" ) with notes is OK.
Fall 2012
Math 419
Problem Set 2
Due on Thu, Sep 20
1) Find an n n matrix A such that Ax = 3x, for all x Rn .
2) The two column vectors v1 and v2 of a 2 2 matrix A are shown in
the gure. Consider the linear transformation T (x) = Ax. Sketch the
2
).
vecto
Math 419
Fall 2012
Problem Set 3
Due on Thu, Sep 27
1) Find vectors that span the
11
1 1
(a) A =
11
kernel of A.
1
1 ,
(b) A =
1
111
123
.
2) Let P2 denote the set of all polynomials of degree 2. We can write
any polynomial f (x) P2 as f (x) = a + bx + cx
Math 419
Fall 2012
Problem Set 4
Due on Thu, Oct 4
1) Are the following matrices linearly independent?
11
11
2) Find the matrix of
2 2 matrices U 22
1
12
M
0
01
,
12
34
,
23
57
,
14
68
.
the linear transformation T from upper triangular
to U 22 with respe
Math 419
Fall 2012
Problem Set 5
Due on Thu, Oct 11
1) Perform the Gram-Schmidt process
1
1
,
1
1
on the following vectors.
6
4
.
6
4
2) Find the QR factorization of the following matrix.
6
2
3 6 .
2
3
3) Find an orthonormal basis of the kernel of the
Fall 2012
Math 419
Problem Set 7
Due on Thu, Nov 8
1) Consider a permutation =
1234
3142
.
(a) Find sgn( ).
(b) Find a decomposition of into transpositions.
2) Find the determinants of the following matrices.
123
(a) A = 1 1 1 ,
321
45
3 6
(b) B =
2 7
18
Math 419
Fall 2012
Problem Set 8
Due on Thu, Nov 15
1) Let v be an eigenvector of A with associated eigenvalue . Is v an
eigenvector of A1 ? If so, what is the eigenvalues?
2) If v is an eigenvector of both A and B , is v necessarily an eigenvector
of AB
Fall 2012
Math 419
Problem Set 9
Due on Tue, Nov 20
1) Decide if the matrix is diagonalizable or not. If possible, nd an invertible S and a diagonal D such that S 1 AS = D, where A is the matrix
in question.
201
101
20
12
, (c) 0 1 0 , (d) 0 1 0 .
, (b)
(
Math 419
Fall 2012
Problem Set 10
Due on Thu, Nov 29
1) Find an orthonormal eigenbasis.
001
11
, (b) 0 0 1 ,
(a)
11
111
02
2
0 .
(c) 2 1
2 0 1
1
2) Let L from R3 to R3 be the reection about the line spanned by 0 .
2
(a) Find an orthonormal eigenbasis B fo
Math 419
Fall 2012
Problem Set 11
Due on Thu, Dec 6
1) Find the matrix of the quadratic form q (x1 , x2 ) = x1 x2 .
2) Determine the deniteness of the quadratic form q (x1 , x2 ) = 2x2 +
1
6x1 x2 + 4x2 .
2
3) Consider a quadratic form q (x) = x Ax, where