Problems of Vector Space

EX-5 Is the set consisting of all polynomials of degree 2 or less

together with standard polynomial addition and scalar multiplication

form a Vector Space?

Let u, v, w ∈ V and c, d ∈ R, V = set consisting of all polynomials of

degree 2 or less.

Therefore,

Axiom 1

= ( a 2 + b 2 ) x 2 + ( a 1 + b 1 ) x + ( a 0 + b 0 )

u = a 2 x 2 + a 1 x + a 0 , v = b 2 x 2 + b 1 x + b 0 , w = c 2 x 2 + c 1 x + c 0

u + v = ( a 2 x 2 + a 1 x + a 0 ) + (b 2 x 2 + b 1 x + b 0 )

since u + v is a polynomial of degree 2 or less.

∴ u + v ∈ V

Axiom 2 u + v = ( a 2 x 2 + a 1 x + a 0 ) + (b 2 x 2 + b 1 x + b 0 )

= ( a 2 + b 2 ) x 2 + ( a 1 + b 1 ) x + ( a 0 + b 0 )

= (b 2 + a 2 ) x 2 + (b 1 + a 1 ) x + (b 0 + a 0 )

= (b 2 x 2 + b 1 x + b 0 ) + ( a 2 x 2 + a 1 x + a 0 ) = v + u

Axiom 3 u + (v + w ) = (a 2 x 2 + a 1x + a 0 )

+[( b 2 + c 2 )x 2 + (b 1 + c 1) x + (b 0 + c 0 )]

= (a 2 + b 2 + c 2 ) x 2 + (a 1 + b 1 + c 1) x + (a 0 + b 0 + c 0 )

= [(a 2 + b 2 ) x 2 + (a 1 + b 1) x + (a 0 + b 0 )]+ c 2 x 2 + c 1x + c 0

= (u + v ) + w

Axiom 4 Let 0 = 0 x 2 + 0 x + 0. Then,

u + 0 = ( a 2 + 0 ) x 2 + ( a 1 + 0 ) x + ( a 0 + 0 )

= (0 + a 2) x 2 + ( 0 + a 1) x + ( 0 + a 0) = 0 + u

= a 2x 2 +a 1x +a 0 = u

Axiom 5 Let - u = ( - a 2 ) x 2 + ( - a 1 ) x + ( - a 0 ). Then,

u + -u = [a 2 + (- a 2 )]x 2 +[ a 1 + (- a 1 )]x +[ a 0 + (- a 0 )]

= 0x 2 + 0x + 0 = 0

Axiom 6 cu = ( ca 2 ) x 2 + ( ca 1 ) x + ( ca 0 )

Since cu is a polynomial of degree 2 or less, cu ∈ V.

Axiom 7 c( u + v ) = [c (a 2 + b 2 )]x 2 + [c (a 1 + b 1 )]x + [c (a 0 + b 0 )]

= (ca 2 + c b 2 ) x 2 + (ca 1 + c b 1 ) x + (ca 0 + c b 0 )

= cu + cv

Axiom 8 (c + d ) u = [(c + d ) a2 ] x 2 + [(c + d ) a1 ]x + [(c + d ) a0 ]

= [(ca2 ) x 2 + (ca1 ) x + (ca0 )] + [(da2 ) x 2 + (da1 ) x + ( da0 )]

= cu + du

Axiom 9 c( du ) = c [(da 2 ) x 2 + ( da 1 ) x + ( da 0 )]

= c ( da 2 ) x 2 + c ( da 1 ) x + c ( da 0 )

= (cd ) a 2 x 2 + (cd ) a 1 x + (cd ) a 0 = ( cd )u

Axiom 10 1u = 1a2 x 2 + 1a1 x + 1a0 = a 2 x 2 + a 1 x + a 0 = u

Hence, the set V of all polynomials of degree 2 or less is Vector Space.

Ex–6 Show that Pn the set consisting of all polynomials of degree n

or less with the form together with standard polynomial addition

and scalar multiplication is a vector space.

Ex-7 The set V consisting of all real valued continuous functions

defined on the entire real line together with standard addition and

scalar multiplication. Is V a vector space?

Ex-8 Show that Rn the set consisting of ordered n tuples with the

form together with standard addition and scalar multiplication is a

vector space.

EX – 9 V be the set consisting of all integers with standard addition

and scalar multiplication. Is V a vector space?

For u = 1 ∈V , and c = 0. 7 ∈ R ,

cu = 0 . 7 x 1 = 0 . 7 ∈ V

Axiom 6 is not satisfied.

V the set of all integers is not a vector space.

Ex – 10 V be the set consisting of all rationales with standard

addition and scalar multiplication. Is V a vector space?

2 December 2020 8

EX - 11 Consider the set V of all doubles of real numbers (x, y) and define the

operations ⨁ and ☉ by

1. (x, y) ⨁ (x’, y’) = (x + x’, y + y’)

2. c ☉ (x, y) = (cx, y).

Is V a Vector Space under the given operation over the field R?

Let u, v, w ∈ V and c, d ∈ R

u = (x, y), v = (x’, y’) & w = (x’’, y’’)

Axiom 1

u ⨁ v = (x, y) ⨁ (x’, y’) = (x + x’, y + y’) ∈ V

∴ u + v ∈ V

Axiom 2 u ⨁ v = (x, y) ⨁ (x’, y’) = (x + x’, y + y’)

= (x’ + x, y’ + y) = (x’, y’) ⨁ (x, y) = v ⨁ u

Axiom 3 u ⨁ (v ⨁ w) = (x, y) ⨁ [(x’, y’) ⨁ (x’’, y’’)]

= (x, y) ⨁ (x’ + x’’, y’ + y”)

= (x + x’ + x’’, y + y’ + y”)

= (x + x’, y + y’) ⨁ (x” + y”)

= (u ⨁ v) ⨁ w

Axiom 4 u ⨁ 0 = (x, y) ⨁ (0, 0) = (x + 0, y + 0) = (x , y) = u

Axiom 5 u ⨁ (-u) = (x, y) ⨁ (-x, -y) = (x + (-x), y + (-y)) = (0 , 0) = 0

Axiom 6 c ☉ u = c ☉ (x, y) = (cx, y) ∈ V.

Axiom 7 c ☉ ( u ⨁ v ) = c ☉ [(x, y) ⨁ (x’, y’)] = c ☉ (x + x’, y + y’)

= (c (x + x’), y + y’) = (cx + cx’, y + y’)

(c ☉ u) ⨁ (c ☉ v ) = (cx, y) ⨁ (cx’, y’)

= (cx + cx’, y + y’)

∴ c ☉ ( u ⨁ v ) = (c ☉ u) ⨁ (c ☉ v )

Axiom 8 (c ⨁ d ) ☉ u = ((c + d) x, y) = (cx + dx, y)

c ☉ u ⨁ d ☉ u = (cx, y) ⨁ (dx, y) = (cx + dx, 2y)

∴ (c ⨁ d ) ☉ u ≠ c ☉ u ⨁ d ☉ u

∴ Given Set V is not a Vector Space under the defined operations.

EX – 12 V the set consisting of all vectors in R2 with Standard addition and

nonstandard scalar multiplication defined by 𝐜𝒙𝟏𝒙𝟐

=𝒄𝒙𝟏𝟎

. Is V a Vector

Space?

Let 𝐮 =𝒙𝟏𝒙𝟐

∈ 𝑹𝟐,

𝟏 ∙ 𝐮 = 𝟏𝒙𝟏𝒙𝟐

=𝒙𝟏𝟎

≠ 𝒖

Axiom 10 is not satisfied.

Given set V is not a Vector Space under the defined operations.

EX - 6 Consider the set V of all doubles of positive real numbers (x, y) and

define the operations ⨁ and ☉ by

1. (x, y) ⨁ (x’, y’) = (x x’, y y’)

2. c ☉ (x, y) = (𝒙𝒄, 𝒚𝒄).

Is V a Vector Space under the given operation over the field R?

Let u, v, w ∈ V and c, d ∈ R

u = (x, y), v = (x’, y’) & w = (x’’, y’’)

Axiom 1

u ⨁ v = (x, y) ⨁ (x’, y’) = (x x’, y y’) ∈ V

∴ u + v ∈ V

Axiom 2 u ⨁ v = (x, y) ⨁ (x’, y’) = (x x’, y y’)

= (x’ x, y’ y) = (x’, y’) ⨁ (x, y) = v ⨁ u

Axiom 3 u ⨁ (v ⨁ w) = (x, y) ⨁ [(x’, y’) ⨁ (x’’, y’’)]

= (x, y) ⨁ (x’ x’’, y’ y”)

= (x x’ x’’, y y’ y”)

= (x x’, y y’) ⨁ (x” + y”)

= (u ⨁ v) ⨁ w

Axiom 4 u ⨁ 0 = (x, y) ⨁ (1, 1) = (x 1, y 1) = (x , y) = u, 0 = (1, 1) ∈ V

Axiom 5 u ⨁ v = (x, y) ⨁ (1/x, 1/y) = (x (1/x), y (1/y)) = (1, 1) = 0

Axiom 6 c ☉ (x, y) = (𝒙𝒄, 𝒚𝒄) ∈ V.

Axiom 7 c ☉ ( u ⨁ v ) = c ☉ [(x, y) ⨁ (x’, y’)] = c ☉ (x x’, y y’)

= ((𝒙𝒙′)𝒄, (𝒚𝒚′)𝒄)

(c ☉ u) ⨁ (c ☉ v ) = (𝒙𝒄, 𝒚𝒄) ⨁ ((𝒙′)𝒄, (𝒚′)𝒄)

= ((𝒙𝒙′)𝒄, (𝒚𝒚′)𝒄)

∴ c ☉ ( u ⨁ v ) = (c ☉ u) ⨁ (c ☉ v )

Axiom 8 (c ⨁ d ) ☉ u = (c + d) ☉ (x, y) = (𝒙𝒄+𝒅, 𝒚𝒄+𝒅)

c ☉ u ⨁ d ☉ u = (𝒙𝒄, 𝒚𝒄) ⨁ (𝒙𝒅, 𝒚𝒅) = (𝒙𝒄𝒙𝒅, 𝒚𝒄𝒚𝒅) = (𝒙𝒄+𝒅, 𝒚𝒄+𝒅)

∴ (c ⨁ d ) ☉ u = c ☉ u ⨁ d ☉ u

Axiom 9 c ☉ (d ☉ u) = c ☉ (𝒙𝒅, 𝒚𝒅) = ((𝒙𝒅)𝒄, (𝒚𝒅)𝒄) = (𝒙𝒄𝒅, 𝒚𝒄𝒅)

(c ☉ d) ☉ u = cd ☉ (x, y) = (𝒙𝒄𝒅, 𝒚𝒄𝒅)

∴ c ☉ (d ☉ u) = (c ☉ d) ☉ u

Axiom 10 1 ☉ u = 1 ☉ (x, y) = (𝒙𝟏, 𝒚𝟏) = (x, y) = u.

∴ Given Set V is a Vector Space under the defined operations.

EX - 13 Consider the set V of all positive real numbers (𝑹+) and define the

operations ⨁ and ☉ by

1. x ⨁ y = x y

2. c ☉ x = 𝒙𝒄

Is V a Vector Space under the given operation over the field R?

Ex – 14 Consider the set V of all doubles of real numbers

(x, y) and define the operations ⨁ and ☉ by,

(x, y) ⨁ (x’, y’) = (x + x’, 0) & c ☉ (x, y) = (cx, 0).

Is V a Vector Space under the given operation over the field R?

Ex – 15 Show that Every plane passing through the origin is a Vector

Space.

Ex – 16 If a mass m is placed at the end of a spring, and if the mass

is pulled downward and released, the mass spring system will

begin to oscillate. The displacement y of the mass from its resting

position is given by a function of the form 𝒚 = 𝒄𝟏𝒄𝒐𝒔𝒘𝒕 + 𝒄𝟐𝒔𝒊𝒏𝒘𝒕,

where w is a constant. Show that the set of all such displacement

functions is a vector space.