1. This chapter introduces the underlying structure of linear algebra, that of
a finite-dimensional vector
space. The definition of a vector space 𝑉, whose elements are called
vectors, involves an arbitrary field 𝐾,
whose elements are called scalars. The following notation will be used
(unless otherwise stated or
implied):
2.
3. Unit-1: Vector Spaces
• Vector Spaces
DEFINITION: Let V be a nonempty set with two operations:
(i) Vector Addition: This assigns to any 𝑢, 𝑣 ∈ 𝑉 a sum 𝑢 + 𝑣 ∈ 𝑉 .
(ii) Scalar Multiplication: This assigns to any 𝑢 ∈ 𝑉, 𝑘 ∈ 𝐾 a product 𝑘𝑢 ∈ 𝑉
Then 𝑉 is called a vector space (over the field 𝐾) if the following axioms hold for
anyvectors 𝑢, 𝑣, 𝑤 ∈ 𝑉 :
[A1] 𝑢 + 𝑣 + 𝑤 = 𝑢 + (𝑣 + 𝑤)
[A2] There is a vector in 𝑉, denoted by 0 and called the zero vector, such that, for any
𝑢 ∈ 𝑉
𝑢 + 0 = 𝑢 = 0 + 𝑢
[A3] For each 𝑢 ∈ 𝑉; there is a vector in 𝑉, denoted by −𝑢, and called the negative of 𝑢,
such that 𝑢 + (−𝑢) = 0 = 𝑢 + (−𝑢).
[A4] 𝑢 + 𝑣 = 𝑣 + 𝑢
4. [M1] 𝑘(𝑢 + 𝑣) = 𝑘𝑢 + 𝑘𝑣, for any scalar 𝑘 ∈ 𝐾.
[M2] 𝑎 + 𝑏 𝑢 = 𝑎𝑢 + 𝑏𝑢, for any scalars 𝑎, 𝑏 ∈ 𝐾.
[M3] 𝑎𝑏 𝑢 = 𝑎 𝑏𝑢 , for any scalars 𝑎, 𝑏 ∈ 𝐾 .
[M4] 1𝑢 = 𝑢, for the unit scalar 1 ∈ 𝐾
Note:
(a) Any sum 𝑣1 + 𝑣2 + 𝑣3 of vectors requires no parentheses and does
not depend on the order of the summands.
(b) The zero vector 0 is unique,
(c) the negative −𝑢 of a vector 𝑢 is unique.
(d) (Cancellation Law) If 𝑢 + 𝑤 = 𝑣 + 𝑤, then 𝑢 = 𝑣.
5. (a) Any sum 𝑣1 + 𝑣2 + 𝑣3 of vectors requires no
parentheses and does not depend on the order of the
summands.
• (b) The zero vector 0 is unique,
