University Mathematics Handbook (2015)

X. Algebra

Chapter 4. Vector Spaces

4.1  Definition

a.  An ordered set of scalars of field is a vector.

b.  Let be a vector above and a scalar of . A vector multiplied by a scalar is vector .

c.  The sum of vectors and is vector .

d.   is called n-dimensional real (complex) vector space.

e.  Non-empty subset of is called subspace if:

1.  For every pair of vectors and of , vector is also of .

2.  For every scalar and vector , vector .

4.2  Linear Dependency and Independency

a.  Vector , when , , is called a linear combination of vectors .

b.  The set of all linear combinations

is called subspace spanned by or the span of vectors . It is denoted

c.  A set of vectors are linearly dependent if there exist scalars , not all of which are zero, such that

Otherwise, the set is said to be linearly independent, or LI.

d.  Vectors and are linearly dependent if and only if they are proportionate, that is, there exists scalar such that or .

e.  If set of vectors is linearly independent, then every of its subsets is linearly independent.

f.  Set of non-zero vectors (finite or infinite) is linearly dependent if and only if it has a finite subset of linearly dependent vectors.

is linearly independent if it has no finite set of linearly dependent vectors.

4.3  Basis and Dimensions of Vector Spaces

a. A set of vectors is called a basis of subspace if:

1. is linearly independent.

2.

b. The vectors

Are the natural basis of .

c. The number of independent vectors in a subspace is not greater than the number of the vectors spanning it.

d. If and are bases of subspace , then .

e. The number of vectors in a basis of subspace is called the dimension of and is denoted .

A vector space with a dimension is called an -dimensional space.

f.  Any set of less than vectors in an -dimensional space can not span .

g.  Any set of more than vectors in an -dimensional space is linearly dependent.