Math 568
Row Echelon Form and Number of Solutions
1. Row Echelon Form In these notes we will define one of the most important forms of a matrix. It is one of the "easier" forms of a system to solve, i
Mathematics 568 Au, Wi, Sp, Su (1st Term) Prerequisite:
3 cr.
Introductory Linear Algebra I
Mathematics 254. Not open to students with credit for 571. Catalog Description: The n-dimensional Euclidean
Math568 Homework II Solution
Lee, Gangyong
1
Exercises 2.1
1. Solution: x y +
3
5z = 0 is a linear equation.
11. Solution: We have 3x 6y = 0, so x = 2y . Hence, the solution set is
x = 2t
y=t
19. Solu
1
Exercise 1.1
17. Solution:
xa
xa
( x a) 2 x
a x
(a x) + a
x
x
2
=
=
=
=
=
=
=
2(x 2a)
2x 4a
(2x 4a) 2x
4a
4a + a
3a
3a
Exercise 1.2
1
2
3. Solution: u = 2 and v = 3, so we have
3
1
uv = 12+23+31
= 2
Quiz 12
Autumn 2013
Name:
Math 2568
Problem 1 (6 points)
1
1
1 2
0 2 1
2
. Find the eigenvalues of A, and for each eigenvalue of A, nd the algebraic
Let A =
1 1 3
2
0
0 1
0
multiplicity, the geome
Quiz 6
Autumn 2013
Name:
Math 2568
Problem 1 (6 points)
Are the vectors v1 = (0, 1, 1, 1) , v2 = (0, 0, 1, 1) , v3 = (1, 1, 1, 0) linearly independent? For full credit, be sure to
justify your answer
Quiz 5
Name:
Autumn 2013
Math 2568
There are two problems on this quiz; make sure to look at the back.
Problem 1 (6 points)
Find the angle between the vectors a = (2, 1) and b = 3
1
2,
1
2
.
Solution
Quiz 8
Name:
Autumn 2013
Math 2568
Problem 1 (6 points)
Let P be the vector space of degree 3 polynomials. Consider the subset
Q = cfw_p P : 2 p (1) + p (3) = 0.
Is Q a subspace of P ? If so, explain
Quiz 7
Name:
Autumn 2013
Math 2568
Problem 1 (6 points)
1
0 2
1
10
2
1
1 6 7 which is row equivalent to 0 1 12 4 .
Consider the matrix A = 3
4 1
4
8
00
0
0
For full credit, nd a basis for (and dimensi
Math 568
Systems of Linear Equations and Matrices
In these notes, we define a linear system and their associated matrices. We also indicate the algebra which can be preformed on these objects.
1. Defi
Math 568 (05/27/11)
Lecturer : Gangyong Lee
5.3 The Gram-Schmidt Process and
the QR Factorization
The Gram-Schmidt Process
1
2
1 and x2 = 0
Example 5.12 Let W = span(x1 , x2 ), where x1 =
0
1
Constru
Math 568 (05/25/11)
Lecturer : Gangyong Lee
5.2 Orthogonal Complements and
Orthogonal Projections
Orthogonal Complements
Denition Let W be a subspace of Rn . We say that a vector v in Rn is
orthogonal
Math 568 (04/01/11)
Lecturer : Gangyong Lee
1.3 Lines and Planes
Lines in R2 and R3
1. Denition The normal form of the equation of a line l in R2 is
n (x p) = 0 or n x = n p
where p is a specic point
Math 568 (03/30/11)
Lecturer : Gangyong Lee
1.2 Length and Angle: The Dot Product
The Dot Product
1. Denition If u =
u1
u2
.
.
.
and v =
un
v1
v2
.
.
.
,
vn
then the dot product u v of u and v is d
Math 568 (03/28/11)
Lecturer : Gangyong Lee
1.1 The Geometry and Algebra of Vectors
Vectors in the Plane
1. Denition of Vector A vector is a direct line segment that corresponds to
a displacement from
Math568 Homework VIII Solution
Lee, Gangyong
1
Exercises 4.3
13
.
2 6
(a) characteristic polynomial
|A I | = 2 7 + 12
(b) eigenvalues
2 7 + 12 = 0, and we have 1 = 3 and 2 = 4.
(c) a basis for each ei
Math568 Homework VI Solution
Lee,Gangyong
1
Exercises 3.6
1. Solution: A =
2 1
34
1
2
,u =
2 1
34
2 1
34
TA (u) = Au =
TA (v ) = Av =
1
. So the standard matrix is
1
1
1
and T (e3 ) =
00+1
0+03
. So w
Math568 Homework V Solution
Lee, Gangyong
1
Exercises 3.3
11. Solution: The coecient matrix is A =
The inverse of A is A1 =
1
2315
x = A1 b =
3 1
5 2
3 1
5 2
21
.
53
3 1
=
. So the solution is
5 2
1
2
Math 2568 Homework #1 due: 20 Jan 2017 (at beginning of class or beforehand)
All these problems must be done by hand no software package or computer language
can be used.
Include the vertical line in