L Q4 Show all steps for the question to be worth any marks G

L. Q4 Show all steps for the question to be worth any marks.

Given the set of all pairs of real numbers of the form (1, x) with the operations (1, y)+(1, y\') = (1, y + y\') and(1, y) = (1, ky), Cheek if the vector space axioms 1 (closure under addition), 6 (close under multiplication by scalar), 7 (distributive property of scalar multiplication with respect to vector addition), and 8 (distributive property of scalar multiplication with respect to scalar addition) are satisfied.

Solution

For gIven operations, (1,y) + (1,y\') = (1,y+y\')

From definition, y and y\' R.

i) Axiom 1 : Closure under addition

since, both y & y\' are real so the sum of y and y\' will also be real i.e. y+y\' is real.

So, (y+y\') R

Thus given operation satisfies Axiom 1.

ii) Axiom 6 : Closure under multiplication by scalar

As we know Scalar multiplication of real number results to real number.

let\'s assume that c R

Thus, c(y+y\') R

Thus given operation satisfies Axiom 6.

iii) Axiom 7 : Distributive property of scalar multiplication with respect to vector addition.

(1,cy) + (1,cy\') = (1,cy+cy\') = (1,c(y+y\'))

Thus given operation satisfies Axiom 7.

iv) Axiom 8: Distributive property of scalar multiplication with respect to scalar addition.

(1,(c+d)y) + (1,(c+d)y\') = (1,(c+d)y+(c+d)y\') = (1,c(y+y\')+d(y+y\'))

Thus given operation satisfies Axiom 8.

For gIven operations, k(1,y) = (1,ky)

From definition, y R.

i) Axiom 1 : Closure under addition

since, y and ky are real so the sum of y and ky will also be real i.e. y+ky is real.

So, (y+ky) R

Thus given operation satisfies Axiom 1.

ii) Axiom 6 : Closure under multiplication by scalar

As we know Scalar multiplication of real number results to real number.

let\'s assume that c R

Thus, c(ky) R

Thus given operation satisfies Axiom 6.

iii) Axiom 7 : Distributive property of scalar multiplication with respect to vector addition.

k(1,cy) = (1,kcy)

Thus given operation satisfies Axiom 7.

iv) Axiom 8: Distributive property of scalar multiplication with respect to scalar addition.

k(1,(c+d)y) = (1,kcy+kdy)

Thus given operation satisfies Axiom 8.