Group Statistics
Therapy
Depression
N
Mean
Std. Deviation
Std. Error Mean
1.00
15
56.0000
9.41883
2.43193
2.00
15
45.0000
7.63451
1.97122
Independent Samples Test
Levene's Test for
Equality of Variances
t-test for Equality of Means
95% Confidence
F
Depres
Correlations
Correlations
Emission Million
GDP Trillion of $
GDP Trillion of $
Pearson Correlation
of Metric tons
1
Sig. (2-tailed)
N
Emission Million of Metric tons
Pearson Correlation
Sig. (2-tailed)
N
.882*
.001
10
10
*
1
.882
.001(P-Value)
10
10
*. Co
Q-1)
a. Mean, Median and Mode
Statistics
Credit
N
Valid
15
Missing 0
Mean
17.4000
Median
18.0000
Mode
18.00
The above table show that the credit hours taken during the final term of senior year has a mean
of 17.4, a media of 18, and a mode with equal to t
EXERCISE I
(A & B)
Frequencies
Statistics
Movies
N
Valid
25
25
0
0
1.4800
1.0000
1.12250
.00000
Missing
Mean
Std. Deviation
Frequency
The Sample has a mean of 1.48 and a standard deviation of 1.12
Frequency Table
Movies
Cumulative
Frequency
Valid
Percent
Material zur der Prfung im Fach Biologie
1)
Empfohlene Literatur
Duden: Basiswissen Schule, Biologie-Abitur, Duden-Verlag
Linder: Biologie, Schroedel-Verlag
Natura: Biologie fr Gymnasien, 3b NRW, Klett-Verlag
2)
Struktur der Klausur
Die Dauer der Klausur
NOTE to prospective students: This syllabus is intended to provide students
who are considering taking this course an idea of what they will be learning. A
more detailed syllabus will be available on the course site for enrolled students
and may be more c
Mathematics IA
Worked Examples
ALGEBRA: THE VECTOR SPACE Rn
Produced by the Maths Learning Centre,
The University of Adelaide.
May 1, 2013
The questions on this page have worked solutions and links to videos on
the following pages. Click on the link with
MATH 423
Linear Algebra II
Lecture 8:
Subspaces and linear transformations.
Basis and coordinates.
Matrix of a linear transformation.
Linear transformation
Denition. Given vector spaces V1 and V2 over a
eld F, a mapping L : V1 V2 is linear if
L(x + y) = L
Worksheet 7: Linear transformations and
matrix multiplication
14. Use the denition of a linear transformation to verify whether the
given transformation T is linear. If T is linear, nd the matrix A such that
T (x) = Ax for each vector x.
T (x1 ) = |x1 |;
Worksheet 14: Dimension and linear
transformations
1. Lay, 4.5.13.
Answer: The dimension of Col A is 3, the dimension of Nul A is 2.
2. Lay, 4.5.19.
Answers: (a) True (b) False (does not need to pass through the origin)
(c) False (the dimension is 5, as a
Subspaces
Sinan Ozdemir, Section 9
I did not get to make it to subspaces today in class, so I decided to make this study sheet for you guys to briey
discuss Sub Spaces.
1
Introduction
We all know what Vector Spaces are (ie. R, R2 , R3 , etc) and we also k
Worksheet 11: Subspaces
We will consider the following vector spaces:
Rn , the spaces we studied before;
Pn , the space of all polynomials in one variable of degree n;
P, the space of all polynomials.
14. Are the following sets subspaces of R2 ?
(1) Th
Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet - Answers
Gaussian Elimination
February 5, 2010
Solve each of the following systems of linear equations, using the technique
of Gaussian elimination.
A.
3x + 2y + 3z 2w = 1
x+y+
Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Gaussian Elimination
Worksheet
February 5, 2010
Solve each of the following systems of linear equations, using the technique
of Gaussian elimination.
A.
3x + 2y + 3z 2w = 1
x+y+z =3
x + 2
Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet
Vector Spaces, Basis & Dimension
March 3, 2010
A. For each of the following sets of n-vectors, decide whether or not it is a real
vector space in Rn . If it is, nd a basis for i
Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet - Answers
Linear Independence and Span
March 1, 2010
A. For each of the following sets of vectors in R3 , determine whether it is linearly
independent, and describe its span. If
Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet - Answers
Inverses
February 26, 2010
A. Let A and B be invertible n n matrices. Use the denition of the inverse,
and any facts that you know about it, to show that
1. A
1
= A1 ,
Test 1 Review Solution
Math 342
(1) Determine whether cfw_(x, y, z) R3 : x + y + z = 1 is a subspace of R3 or not.
Solution: 0 + 0 + 0 = 1. Additive identity is not in the set so not a subspace.
(2) Is the dimension of Pm (F ) is m ? Why ?
Solution: Dimen
Mathematics 109, Linear Algebra
Winter 2006
Assignment 1 Solutions
1. Prove or disprove if the following sets are subspaces of R3 .
(a) The set S = cfw_(a, 0, 0)| a R is a subspace of R3 .
Proof: S is non-empty since 0 = (0, 0, 0) S.
The set S is closed u
Area Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project X
BACKGROUND: Congruent and Similar triangles, Squares and Parallelograms, Coordinate Transformations and Images.
In this project we introduce the notion of area with four new
Circles Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project X
BACKGROUND: General Axioms, Half planes, Isosceles Triangles, Perpendicular Lines, Congruent Triangles, Parallel Postulate and Similar Triangles.
DEFN: A circle of radius
GEOMETRY
The University of the State of New York
REGENTS HIGH SCHOOL EXAMINATION
GEOMETRY
Thursday, January 28, 20109:15 a.m. to 12:15 p.m., only
Student Name: _
School Name: _
Print your name and the name of your school on the lines above. Then turn to
t
Coordinate Plane Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project XI
BACKGROUND: Euclidean Axioms, Half Planes, Unique Perpendicular Lines, Congruent
and Similar Triangle Theorems, Parallelograms.
PERPENDICULAR PERPENDICULAR TO PA
Coordinate Transformations Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project X
BACKGROUND: General Axioms, Congruent and Similar Triangles, Coordinate Plane,
Distances Between Points in the Plane, Lines Through the Origin.
DEFINITI
Isometry Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project IX, Part II
BACKGROUND: Euclidean Axioms, Midpoints, Perpendicular Bisectors, Congruent Triangles,
Symmetry, Maps, Bijections.
DEFN: A Euclidean isometry is a map, F : E 2
Symmetry Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project IX, Part I
BACKGROUND: Euclidean Axioms, Midpoints, Perpendicular Bisectors, Congruent Triangles.
LINE SYMMETRY: Plot points A = (2, 3) A = (2, 3) B = (5, 0), B = (5, 0) ,
Parallel Postulate Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009
BACKGROUND: All axioms of Absolute Geometry and the Theorems of Congruent
Triangles. In this lesson, all points and lines will lie in a common plane.
DEFINITION:Two lines
Similar Triangles Lesson and Project
C. Sormani, MTTI, Lehman College, CUNY
MAT631, Fall 2009, Project VII
BACKGROUND: Euclidean geometry axioms including the parallel postulate and the
SSS, SAS, ASA, Vertical Angle, Alternate Interior Angles and Parallel