Handout OC - Orthogonal Complements
Recall the denition of dot product (sometimes called the inner product) in Rn :
(a1 , a2 , , an ) (b1 , b2 , , bn ) = a1 b1 + a2 b2 + an bn
Two vectors (a1 , a2 , , an ), (b1 , b2 , , bn ) Rn are said to be orthogonal t
Handout F - Fibonacci type sequences
Elementary Linear Algebra Class
Example: Given a0 = 2, a1 = 3, ak+1 = 3ak 2ak1 . Find the closed form formula for the term
ak .
The rst 5 terms of the sequence are: a0 = 2, a1 = 3, a2 = 5, a3 = 9, a4 = 17
For k 0, dene
Handout EE - Eigenvalues/Eigenvectors
This section is a preview of eigenvaluesone of the most important concepts in
linear algebra. They are used in many, many applications especially in physics
and engineering. For example, they have applications in the
George Hukles Problem
A Reverse Engineering Problem
For n N,
n
i=1
n2 n
n(n + 1)
=
+ ,
i=
2
2
2
n
i2 =
n(n + 1)(2n + 1)
n3 n2 n
=
+
+ , and
6
3
2
6
i3 =
n2 (n + 1)2
n4 n3 n2
=
+
+ .
4
4
2
4
i=1
n
i=1
n
i4 ? The pattern appears to be:
What is the formula f
Graphing Calculator Workshop
Marian K. Hukle, [email protected]; Amy Kim, [email protected]; Chris Valle, [email protected]
POWER ON/OFF
ON to Press turn on calculator. Press 2nd OFF turn off calculator. to
SCREEN CONTRAST
Press 1 7
Press 2nd to m
Math 291 - Practice Problems for Final
These problems are in addition to the Practice Problems for Tests 1 and 2.
1. Find the eigenvalues for each of the following matrices:
A=
a
a
2
a
2
0
2. For what values of a does the matrix A =
B=
2 sin sin
sin
0
0
Math 291 - Practice Problems for Test 2
SHOW AND/OR EXPLAIN ALL YOUR WORK INCLUDING INTERMEDIATE MATRICES. UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT.
1. Are the vectors w1 = (1, 1, 2, 1), w2 = (1, 2, 1, 1) and w3 = (1, 1, 4, 1) linearly independent?
Math 291 - Practice Problems for Test 1
UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT.
3 4
5
5 2 and b = 1. Show that b Spancfw_c1 , c2 by
1. Let c1 and c2 be the columns of A =
6 6
6
writing b as a linear combination of c1 and c2 .
2. Find the value
Matrices on the Graphing Calculator
Entering a Matrix with the TI-83 or TI-84 Plus matrix editor
.
2nd
MATRX, EDIT, and
To enter a matrix on the TI-83, press MATRX On the TI-84, go to
select the matrix you would like to edit. Enter the number of rows the
FIBONACCI TYPE PROBLEM
Given a0 = 1, a1 = 1, ak+1 = 3ak + 4ak1 . Use matrices to nd a closed form formula for ak .
ak
1
for k 0. This means x0 =
.
ak+1
1
Let xk =
xk+1 =
Diagonalize the matrix A =
det(I A) =
0 1
4 3
ak
ak+1
0 1
.
4 3
1
= ( 3) 4 = 2 3 4 =
FIBONACCI TYPE PROBLEM
Given a0 = 0, a1 = 1, ak+1 = 2ak + 3ak1 . Use matrices to nd a closed form formula for ak .
This means x0 =
0
ak
and xk =
for k 0.
1
ak+1
xk+1 =
ak+1
0 1
=
ak+2
3 2
ak
ak+1
Diagonalize the matrix:
det(I A) =
1
= ( 2) 3 = 2 2 3 = ( 3