(1 point Let V =R2 and let H be the subset of V of all points on the Ine-4x + 3y

Question

(1 point Let V =R2 and let H be the subset of V of all points on the Ine-4x + 3y--12. Is H a subspace of the vector space V? 1. Is H nonempty? choose 2. Is H closed under addition? if it is, enter CLOSED. tf it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as . 3 is H closed under scalar muitiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, . 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. choose

Explanation / Answer

1.

Yes.

Set, x=0 in the equation gives, y=-4

So, (0,-4) is in the set H

2.

Set, y=0 and we get x=3 so (3,0) is in the set

Adding, (3,0),(0,-4) which are both in H we get

(3,-4)

-4*3+3*(-4)=-24

Hence sum is not in H so not closed

(3,0),(0,-4)

3.

(0,-4) is in the set

Multiply by scalar -1 we get (0,4)

-4*0+3*4=12

Hence not closed under scalar multiplication

-1,(0,4)

4.

Not a subspace as not closed under addition and scalar multiplication

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