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 thi
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 .
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
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 transfor
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
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 o
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 o
Fall 2012
Math 419
Problem Set 1
Due on Thu, Sep 13
1) Solve the following systems using Gauss-Jordan elimination.
(a)
3x + 4y z = 8
,
6x + 8y 2z = 3
x1 7x2
(b)
(c)
x3
+ x5 = 3
2x5 = 2 ,
x4 + x5 = 1
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
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 c
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 tr
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 .
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 m
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 o
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 qu
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 s
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 + 4x