Vector Spaces: Basic Example Problems

VECTOR SPACES

BASIC EXAMPLE PROBLEMS

Example 1

The set of all n−tuples with entries from a field F is denoted by Fⁿ. This set is a vector space over F with the operations of coordinate wise addition and scalar multiplication.

Solution:

(ie) If u= (a₁, a₂, a₃... a_n) ∈ Fⁿ

v = (b₁, b₂, b₃,... b_n) ∈ Fⁿ and C ∈ F then

 u+v = (a₁, a₂ ... a_n) + (b₁, b₂ ... b_n) = (a₁ + b₁, a₂ + b₂, ... a_n + b_n)

 cu = c(a₁, a₂... a_n) = (ca₁, ca₂, ca₃, ... ca_n)

Thus R³ is a vector space over R.

Here (3,−2,1) + (2,− 5, 2) = (3 + 2, −2−5, 1+2) = (5, −7, 3) and −4 (2, 4, 0) = (−8, 16, 0)

Similarly, C² is a vector space over C.

Let (2+i,3)+(2−4i, 2i) = (4−3i, 3+2i) and

 i(2 − i, 4) = (2i – i², 4i) = (2i + i², 4i) = (1 + 2i, 4i)

Example 2

The set of all m×n matrices with entries from a field F is a vector space, which is denoted by M_m×n (F), with the following operations of matrix addition and scalar multiplication.

Solution:

For A, B ∈ M_m×n (F) and c ∈ F

(A+B)_ij = A_ij+B_ij and (CA)_ij =cA_ij, for 1 ≤ i ≤ m and 1≤j≤n.

Example 3

Let S be any non−empty set and F be any field, and let F (S, F) denote the set of all functions, from S to F. Two functions ƒ and g in F (S, F) are called equal if ƒ (s) = g(s) for each s ∈ S. The set F (S, F) is a vector space with the operations of addition and scalar multiplication defined for f,g ∈ F (S, F) and c ∈ F by (f+g) (s) = f (s) + g(s) and (cf) (s) = c [ƒ (s)] for each s ∈ S.

Example 4

A polynomial with coefficients from a field "F" is an expression of the form.

 ƒ(x) = a_nxⁿ + a_n−1 x^n-1 + a_n−2 x^n-2 + …. + a₁x + a₀

where n is a non−negative integer and each a_k, called the coefficient of x^k is in F. If f(x)=0, if a_n=a_n-1 = a_n−2 = a_n−3 = … a₀ = 0, then x f(x) is called the zero polynomial and its degree is defined to be ‒1 otherwise, the degree of a polynomial is defined to be the largest exponent of x that appears in the representation.

ƒ(x) = a_nxⁿ + a_n−1 x^n-1 + a_n−2 x^n-2 + …. + a₁x + a₀ with a non−zero coefficient.

Two polynomials

ƒ(x) = a_nxⁿ + a_n−1 x^n-1 + a_n−2 x^n-2 + …. + a₁x + a₀ and

 g(x) = b_mx^m + b_m−1 x^m-1 + b_m−2 x^m-2 + …. + b₁x + b₀

 are called equal if m=n and a_i = b_i; for i=0, 1, 2 ... n.

Example 5

Let ƒ(x) = a_nxⁿ + a_n−1 x^n-1 + a_n−2 x^n-2 + …. + a₁x + a₀ and g(x) = b_mx^m + b_m−1 x^m-1 + b_m−2 x^m-2 + …. + b₁x + b₀ be polynomials with coefficients from a field F. Suppose that m≤n and define b_{m +1}=b_{m +2}=b_{m + 3} = b_n = 0.

Then g(x) can be written as

 g(x) = b_nxⁿ + b_n−1 x^n-1 + b_n−2 x^n-2 + …. + b₁x + b₀

Define

 f(x) + g(x) = (a_n+b_n)xⁿ + (a_n-1+b_n−1)x^n-1 + (a_n-2+b_n−2)x^n-2 + …. + (a₁+b₁)x + (a₀+b₀),

and for any c ∈ F, define

 cƒ(x) = ca_nxⁿ + ca_n−1 x^n-1 + ca_n−2 x^n-2 + …. + ca₁x + ca₀

With these operations of addition and scalar multiplication, the set of all polynomials with coefficients from F is a vector space, which is denoted by P (F).

Example 6

Let F be any field. A sequence in F is a function σ from the positive integers into F. The sequence such that σ (n) = a_n for n = 1, 2, ... is denoted by (a_n). Let V consists of all the sequences (a_n) in F. For (a_n) and (b_n) in V and t ∈ F, define (a_n) + (b_n) = (a_n+b_n) and t(a_n) = (ta_n), with these operations V is a vector space.

Example 7

Let S = {(a₁, a₂); a₁, a₂ ∈ R}. For (a₁, a₂), (b₁, b₂) ∈ S and C ∈ R, define (a₁, a₂) + (b₁, b₂) = (a₁ + b₁, a₂+b₂) and C(a₁, a₂) = (Ca₁, Ca₂)

Since the properties

1. x+y = y+x for all x, y ∈ V

2. (x+y) + z = x + (y + z) for all x, y, z ∈ V

3. (a+b) x = ax + bx for a, b ∈ F and x ∈ V.

fail to hold, S is not a vector space with these operations.

Example 8

Let S = {(a₁, a₂) ; a₁, a₂ ∈ R ).

For (a₁, a₂), (b₁, b₂) ∈ S and c ∈ R, define

 (a₁, a₂)+(b₁, b₂) = (a₁+b₁, 0) and

 c (a₁, a₂) = (ca₁, 0).

Then S is not a vector space with these operations since properties 3, 4, and 5 fail.