Lecture 1m
The Scalar Equation of a Hyperplane
(pages 34-36)
In keeping with the trend for this chapter, we will develop the general scalar
equation of a hyperplane by rst focusing our attention on the equation of a
hyperplane in R3 . As I mentioned in Le
Lecture 1s
Finding the Line of Intersection of Two Planes
(page 55)
Now suppose we were looking at two planes P1 and P2 , with normal vectors
n1 and n2 . We saw earlier that two planes were parallel (or the same) if and
only if their normal vectors were s
gamma Concepts of sustainability
: ~ 1 - - ° - question that then arises
L f T ' future generations? The
L do we look after those i
tion itself involves two
tion of current policy obj
is: what are the interests of
next ques
REPORT OF THE COMMITTEE ON
ROADMAP FOR FISCAL CONSOLIDATION
Vijay L Kelkar - Chairman
Indira Rajaraman - Member
Sanjiv Misra - Member
SEPTEMBER 2012
FOREWORD
This Committee was mandated by the Finance Minister to give a report outlining a roadmap
for fisc
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SPECIAL ARTICLE
Financial Sector Reforms: Realities and Myths
R H Patil
Does Indias strategy for financial sector reform need to
be reviewed in the light of what happened in the major
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hitherto the ro
Special articles
Indias Agricultural Development Policy
Pressures for liberalisation and globalisation of Indian agriculture are growing. The focus
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restrictions on both intern
Lecture 1r
Finding the Normal to a Plane
(pages 54-55)
Another feature of the cross product u v is that it is orthogonal to both u and
v. So, if u and v are linearly independent direction vectors for a plane, then
u v is a normal vector for the plane. (Th
Lecture 1a
Vectors in R2
(pages 1-4)
We begin our study of linear algebra by introducing a new concept: vectors.
And we will start our study of vectors by looking at points in the plane.
x1
, where x1 and x2 are
x2
real numbers called the components of th
Lecture 1c
Directed Line Segments
(pages 7-8 )
So far, we have always had our vectors start at the origin, and end at
the point corresponding to our vector. But if we are thinking of vectors as a
direction instead of as a point, then it shouldnt really ma
Lecture 1g
Subspaces
(pages 16-17)
I will begin our discussion of subspaces with the following denition of a subspace, which has been modied from the one given in the text.
Denition A subset S of Rn is called a subspace of Rn if the following conditions
h
Lecture 1f
Vectors in Rn
(pages 14-16)
So far weve been easing you into the world of vectors, but at long last it is time
to proceed to the general world of Rn , for a n a positive integer.
x1
.
Denition Rn is the set of all vectors of the form . , wher
Lecture 1k
Length and Dot Product in R2 and R3
(pages 28-31)
If we go back to visualizing a vector as a directed line segment from the origin to
the point aliated with the vector, then we can compute the length of a vector
as follows:
x1
x2
Denition If x
Lecture 1l
Length and Dot Product in Rn
(pages 31-34)
Okay, now that we eased into the notion of length and the dot product in R2
and R3 , we can expand our denitions to a general Rn .
y1
x1
.
.
Denition Let x = . and y = . be vectors in Rn . Then the
Lecture 1i
Linear Independence and Basis
(pages p.20-23)
The idea that a spanning set may contain unnecessary vectors leads to the
following denition.
Denition A set of vectors cfw_v1 , . . . , vk is said to be linearly dependent if
there exists coecient
Lecture 1e
Vector Equation of a Line in R3
(page 11)
One of the advantages of vectors lies in the lack of changes as we change from R2
to R3 , and eventually to general Rn . This is because we take all the information
and pack it into a single vector. In
Lecture 1d
Vectors in R3
(pages 9-11)
Well, R2 is nice, but theres more to explore in this world. Like a third dimension, for example! And all of the work that weve done so far easily extends to
R3 . (And to general Rn , but well be there soon enough.)
x1
Lecture 1b
Vector Equation of a Line in R2
(pages 5-6)
The equation x2 = (0.5)x1 + 1 denes the x2 component of a point on the line
in terms the x1 component. But when we graph the line, we think less of this
equation, and more of the following two facts:
MATH 106
MODULE 1 LECTURE a COURSE SLIDES
(Last Updated: April 15, 2013)
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University of Waterloo and others
MATH 106
MODULE 1 LECTURE a COURSE SLIDES
(Last Updated: April 15, 2013)
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University of Waterloo and others
MATH 106
MODULE