1S11 (McLoughlin) Tutorial sheet 4
[Week 4, 2013]
Solutions
1. Find parametric equations for the line of intersection of the two planes
2x 4y + 3z = 1
2x + 2y + 2z = 3
Solution: We need a vector parallel to the line (and also a point on the line) to write
Vectors
MA1S1
Tristan McLoughlin
tristan@maths.tcd.ie
Vectors
Some quantities (which we will call scalars) have purely numerical values
(like mass, volume, temperature) while others (which are called vectors)
also have a direction associated with them.
Ex
1S11 (McLoughlin) Tutorial sheet 9
[Week 11, 2012]
1.
(a) [1 point] Is there a matrix that is both strictly upper and strictly lower triangular?
(Explain why not or else give an example of one.)
Solution: The zero matrix of any size, such as the 2 2 matri
MA1s1-tut10 - Sage
07/12/2013 13:34
MA1s1-tut10
Problem 1
i) [2 points] We first define the variables for our system of linear equations.
var('x1,x2,x3,x4')
(x1, x2, x3, x4)
We define a system of equations
eq1=
eq2=
eq3=
eq4=
5*x1-2*x2+x3-4*x4=-3;
2*x1+3*
1S11 (McLoughlin) Tutorial sheet 2
[Week 3, 2013]
Name: Solutions
1.
(a) Show (on the graph) the point P with coordinates (1, 3, 2) and the point Q with
coordinates (3, 2, 4).
(b) Sketch the position vectors of the two points (P for P and Q for Q).
(c) Ca
1S11 (McLoughlin) Tutorial sheet 3
[Week 4 , 2013]
Name: Solutions
1. Find the equation of the plane in space passing through the point (1, 2, 3) perpendicular
(normal) to the vector 6i 5j + 4k.
Solution: We know the equation has the form 6x 5y + 4z = con
1S11 (McLoughlin) Tutorial Solution sheet 6
[Week 8, 2013]
1. Find all solutions of the following system of linear equations by using Gauss-Jordan elimination. (Follow the method exactly.)
4x1 + 2x2 + 22x3 16x4 =
14
6x1 + 7x2 43x3 24x4 = 11
Solution: (3 m
MA1S11 (McLoughlin) Tutorial/Exercise sheet 1
[due week 2, 2013]
Solutions
Each subquestion is worth 2 marks.
1. Consider the two-dimensional plane with the usual coordinate axes x and y
(a) Show (on the graph)
the point P with coordinates (1, 4) and the
Mathematics for Scientists
MA1S1
Tristan McLoughlin
tristan@maths.tcd.ie
MA1S11
This course will cover various aspects of mathematics with a particular
focus on the tools, techniques and methods that are necessary for
understanding topics in your science
Vectors IV
MA1S1
Tristan McLoughlin
October 12, 2013
Anton & Rorres: Ch 3.4, 3.5
Heeron: Ch One, sec II.1 and II.2
Cross products
Denition: The cross product v w of two vectors
v
=
v1 i + v2 j + v3 k,
w
=
w1 i + w2 j + w3 k,
is
v w = (v2 w3 v3 w2 )i + (v3
Vectors IV
MA1S1
Tristan McLoughlin
October 16, 2013
Anton & Rorres: Ch 3.4, 3.5
Heeron: Ch One, sec II.1 and II.2
Cross products
We previously described the the scalar product which takes two vectors and
forms a number or a scalar:
(v, w) V V v w R .
In
Matrices VI-VII
MA1S1
Tristan McLoughlin
November 21, 2013
Anton & Rorres: Ch 1.5, 1.6, 1.7, 10.6
Special matrices
There are matrices that have a special form that makes calculations with
them much easier than the same calculations are as a rule.
Diagonal
Graph Theory
MA1S1
Tristan McLoughlin
November 27, 2013
Anton & Rorres: 10.6
Robin J. Wilson: Introduction to Graph theory
Graph Theory
Recall some denitions (actually we will be a bit more general than before):
A graph consists of points called vertices
Number Systems
MA1S1
Tristan McLoughlin
November 27, 2013
http:/en.wikipedia.org/wiki/Binary numeral system
http:/accu.org/index.php/articles/1558
http:/www.binaryconvert.com
http:/en.wikipedia.org/wiki/ASCII
Counting
Normally we use decimal or base 10 wh
Matrices V
MA1S1
Tristan McLoughlin
November 15, 2013
Anton & Rorres: Ch 1.3, 1.4, 1.5
Elementary matrices
We now make a link between elementary row operations and matrix
multiplication.
Recall now the 3 types of elementary row operations:
(i) multiply al
Matrices - Sage
31/10/2013 21:06
Matrices
Dealing with matrices in SAGE is straightforward. We can
define a matrix with the Matrix() command and then specify the
entries as a list of of rows.
To create the 3 4 matrix
1 2
0 9
3 4
4
8
7
2
12
2
we enter a l
Linear Algebra II & III
MA1S1
Tristan McLoughlin
October 24, 2013
Anton & Rorres: Ch 1.1, 1.2, 1.9
Heeron: Ch One, sec I.1, sec III.3
Solving systems of equations, Gaussian elimination
We will now go over the same ground again but using a new and more
con
Linear Algebra I
MA1S1
Tristan McLoughlin
October 18, 2013
Anton & Rorres: Ch 1.1, 1.2
Heeron: Ch One, sec I.1
What is linear and not linear?
Here are some examples of equations that are plausibly interesting from
some practical points of view
x2 + 3x 4
2
Vectors II
MA1S1
Tristan McLoughlin
tristan@maths.tcd.ie
Anton & Rorres: Ch 3, sec 1 and 2
Heeron: Ch One, sec II.1 and II.2
Vectors - Review
We introduced the notion of a vector, a quantity with magnitude and
direction. Two dimensional vectors can be rep
Vectors III
MA1S1
Tristan McLoughlin
October 9, 2013
Anton & Rorres: Ch 3.3
Heeron: Ch One, sec II.1 and II.2
Dot product
The dot product of two 3-dimensional vectors v and w is given in terms
of the components v = v1 i + v2 j + v3 k and w = w1 i + w2 j +
Number Systems III
MA1S1
Tristan McLoughlin
December 4, 2013
http:/en.wikipedia.org/wiki/Binary numeral system
http:/accu.org/index.php/articles/1558
http:/www.binaryconvert.com
http:/en.wikipedia.org/wiki/ASCII
Converting fractions to binary
So far we ha