Linear Algebra  Spring 2016
Some additional exercises to review the exam
In addition to your previous homeworks, midterms, quizzes, worked examples and exercises done
in class, you can do some of the following exercises from Strang. This list is not exha
Visualization Descriptive Geometry
What is descriptive geometry?
Descriptive Geometry is the graphical solution of point, line
and plane problems in space.
The solutions are accomplished by means of the same
principles of orthographic drawing.
Terminolo
Contours
A contour map is a plot plan that shows the elevation (height) of points
on its surface.
A contour map is a twodimensional drawing of a three dimensional
surface.
1
Contour lines are horizontal lines that represent constant elevations from a
h
Auxiliary View Projection Theory
An auxiliary view is an orthographic view that is projected onto any other
plane other that the frontal, horizontal or profile plane.
1
Folding Line Method
The object is suspended in a glass box to the six principal views,
Rules for Dimensioning
1. Dimensioning must be Complete.
User should not have to make assumptions
Only one possible interpretation
2. Avoid over dimensioning do not add extra dimensions
3. Dimensions should be placed in the view that must clearly descri
Summary
Chapter 1 Principles of Projections
How objects are formed
Orthographic projection (3rd angle)
Choosing Front and Side views
Another method of Orthographic Projection (1 st angle)
Projection Between orthographic views
Curved surfaces
Pictoria
Properties of PLANES
A Plane is a flat surface containing a straight line, that in any position, will
connect two points that lie on the surface
A Plane is positioned by locating three points on the plane that are not in a
straight line.
The edge view mus
Scales
Scales are used to measure distances on technical drawings.
With the advent of computers all drawings are created full size. However,
They cannot be plotted full size, a reduction scale is specified by
the user.
1 unit drawn on paper = a larger nu
Sketching
Freehand technique
Sketching is a thinking process as much as a method of communication.
Quick and rapid  Used to visualize and develop ideas on paper.
The sketch does not have to be an exact representation but is should show
all the necess
Dimensioning
Once a part is defined with an orthographic drawing, the size
information is added in the form of dimensions.
A fully defined part has three elements:
1. Graphics
2. Dimensions
3. Words (notes).
Standards
provide uniform methods of dimensio
Intersections
Intersecting and nonintersecting lines
Lines that intersect have a point of intersection that lies on both
lines and is common to both.
In the above figure (a) the lines intersect because the point of
intersection O, projects as a point of
Solutions to Exercises
64
34 If AB D BA, then B has the same eigenvectors .1; 0/ and .0; 1/ as A. So B is also
diagonal b D !c D 0.
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Math 111 L01
Assignment #5 Solutions
Section A: Problems from Textbook
7.4, page 541
7.5, page 547
Section B: Assorted Problems
1. Let u cos x , then du sin x dx du sin x dx and
sin x
5cos x cos
2
x
dx
1
1
( du )
du
2
5u u
5u u 2
1
A
B
A(5 u ) Bu
1
Math 111 L01
Assignment #1 Solutions
Section A: Problems from Textbook
Section 6.1, Page 406
9. The graph of () = 2 3 is a line with slope 2. It passes the Horizontal Line Test, so is
onetoone.
2
3 2
3
, then 2
,
Algebraic solution: If
2
so is oneto
Working Drawings
Working drawings are a complete set of standardized drawings specifying
the manufacture and assembly of a product based on its design.
The complexity of the design determines the number and types of
drawings. (It is important to give the