MAT2611 - Notes for Linear Algebra (Summary)

MAT2611
Summary Notes




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Notes & Memo

,1 VECTOR SPACES AND SUBSPACES
What is a vector? Many are familiar with the concept of a vector as:

• Something which has magnitude and direction.

• an ordered pair or triple.

• a description for quantities such as Force, velocity and acceleration.

Such vectors belong to the foundation vector space - Rn - of all vector spaces. The
properties of general vector spaces are based on the properties of Rn . It is therefore
helpful to consider briefly the nature of Rn .


1.1 The Vector Space Rn

Definitions

• If n is a positive integer, then an ordered n-tuple is a sequence of n real
numbers (a1 , a2 , . . . , an ). The set of all ordered n-tuples is called n-space and
is denoted by Rn .

When n = 1 each ordered n-tuple consists of one real number, and so R may be
viewed as the set of real numbers. Take n = 2 and one has the set of all 2-tuples
which are more commonly known as ordered pairs. This set has the geometrical
interpretation of describing all points and directed line segments in the Cartesian x−y
plane. The vector space R3 , likewise is the set of ordered triples, which describe all
points and directed line segments in 3-D space.
In the study of 3-space, the symbol (a1 , a2 , a3 ) has two different geometric in-
terpretations: it can be interpreted as a point, in which case a1 , a2 and a3 are the
coordinates, or it can be interpreted as a vector, in which case a1 , a2 and a3 are
the components. It follows, therefore, that an ordered n-tuple (a1 , a2 , . . . , an ) can be



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,viewed as a “generalized point” or a “generalized vector” - the distinction is math-
ematically unimportant. Thus, we can describe the 5-tuple (1, 2, 3, 4, 5) either as a
point or a vector in R5 .
Definitions

• Two vectors u = (u1 , u2 , . . . , un ) and v = (v1 , v2 , . . . , vn ) in Rn are called equal
if
u1 = v1 , u2 = v2 , . . . , un = vn

• The sum u + v is defined by

u + v = (u1 + v1 , u2 + v2 , . . . , un + vn )

• Let k be any scalar, then the scalar multiple ku is defined by

ku = (ku1 , ku2 , . . . , kun )

• These two operations of addition and scalar multiplication are called the stan-
dard operations on Rn .

• The zero vector in Rn is denoted by 0 and is defined to be the vector

0 = (0, 0, . . . , 0)

• The negative (or additive inverse) of u is denoted by -u and is defined by

−u = (−u1 , −u2 , . . . , −un )

• The difference of vectors in Rn is defined by

v − u = v + (−u)

The most important arithmetic properties of addition and scalar multiplication
of vectors in Rn are listed in the following theorem. This theorem enabes us to
manipulate vectors in Rn without expressing the vectors in terms of componenets.

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, Theorem 1.1. If u = (u1 , u2 , . . . , un ), v = (v1 , v2 , . . . , vn ), and w = (w1 , w2 , . . . , wn )
are vectors in Rn and k and l are scalars, then:

1. u + v = v + u

2. u + (v + w) = (u + v) + w

3. u + 0 = 0 + u = u

4. u + (−u) = 0; that is, u − u = 0

5. k(lu) = (kl)u

6. k(u + v) = ku + kv

7. (k + l)u = ku + lu

8. 1u = u


1.2 Generalized Vector Spaces

The time has now come to generalize the concept of a vector. In this section a set of
axioms are stated, which if satisfied by a class of objects, entitles those objects to be
called “vectors”. The axioms were chosen by abstracting the most important prop-
erties (theorem 1.1). of vectors in Rn ; as a consequence, vectors in Rn automatically
satisfy these axioms. Thus, the new concept of a vector, includes many new kinds
of vector without excluding the “common vector”. The new types of vectors include,
among other things, various kinds of matrices and functions.
Definition
A vector space V over a field F is a nonempty set on which two operations are
defined - addition and scalar multiplication. Addition is a rule for associating with
each pair of objects u and v in V an object u + v, and scalar multiplication is a rule
for associating with each scalar k ∈ F and each object u in V an object ku such that


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